On Group-Like Magmoids

TL;DR

This paper explores various generalizations of group-like structures using partial binary operations, introducing new classes of magmoids and establishing foundational results and analogues of classical theorems.

Contribution

It defines and analyzes multiple new classes of magmoids with generalized associativity, identities, and inverses, extending the theory of group-like structures.

Findings

Introduction of poloids, groupoids, and skew variants.

Derivation of basic properties and connections between classes.

Proofs of analogues to classical theorems like Ehresmann-Schein-Nampooribad.

Abstract

A magmoid is a non-empty set with a partial binary operation; group-like magmoids generalize group-like magmas such as semigroups, monoids and groups. In this article, we first consider the many ways in which the notions of associative multiplication, identities and inverses can be generalized when the total binary operation is replaced by a partial binary operation. Poloids, groupoids, skew-poloids, skew-groupoids, prepoloids, pregroupoids, skew-prepoloids and skew-pregroupoids are then defined in terms of generalized associativity, generalized identities and generalized inverses. Some basic results about these magmoids are derived, and connections between poloid-like and prepoloid-like magmoids, in particular semigroups, are described. Notably, analogues of the Ehresmann-Schein-Nampooribad theorem are proved.

Equations

$$\mathopen{}\mathclose{{}\left(\rho,X_{1},\ldots,X_{n}}\right),$$

$$\mathopen{}\mathclose{{}\left\{x_{i}\mid\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{i},\ldots,x_{n}}\right)\in\rho}\right\},$$

$$\mathopen{}\mathclose{{}\left(\rho,X,Y}\right)$$

$$\mathopen{}\mathclose{{}\left(\phi,X,Y}\right)$$

$$\mathfrak{m}:X\times X\rightarrow X,\qquad\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathfrak{m}\mathopen{}\mathclose{{}\left(x,y}\right)=:xy.$$

$$\mathopen{}\mathclose{{}\left(\mu,X,X,X}\right)$$

$$\begin{array}[]{lc}\mathrm{if}\;p\;\mathrm{is\>false\>then}&p\wedge q\;\mathrm{is}\>\mathrm{false;}\\ \mathrm{else\;if}\;q\;\mathrm{is\>false\>then}&p\wedge q\;\mathrm{is}\>\mathrm{false;}\\ \mathrm{else}&p\wedge q\;\mathrm{is}\;\mathrm{true.}\end{array}$$

$${\pi }_{x_1,\dots ,x_n}={\pi }_{x_1,\dots ,x_n}^{\prime }.$$

$$\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(zu}\right)}\right)}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(yz}\right)u}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)u}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)u}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)\mathopen{}\mathclose{{}\left(zu}\right)}\right).$$

$$\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)\rightarrow\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right),\quad\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)\wedge\mathopen{}\mathclose{{}\left(yz}\right)}\right)\rightarrow\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right),$$

$$\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)\rightarrow\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right).$$

$$\boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)}=\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}}\right),\quad\boldsymbol{\uprho}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)}=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}x^{-1}}\right)$$

$$\displaystyle\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)x^{-1}}\right)=\mathopen{}\mathclose{{}\left(xx^{-1}}\right),$$

$$\displaystyle\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)x^{-1}}\right)x}\right)=\mathopen{}\mathclose{{}\left(x^{-1}x}\right),$$

$$\displaystyle\mathrm{(resp.\quad}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)x^{-1}}\right)x}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)}\right)=\mathopen{}\mathclose{{}\left(x^{-1}x}\right),$$

$$\displaystyle\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)x^{-1}}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)}\right)=\mathopen{}\mathclose{{}\left(xx^{-1}}\right)\mathrm{\>).}$$

$$*:P\rightarrow P,\qquad x\mapsto*\mathopen{}\mathclose{{}\left(x}\right)=:x^{*}$$

$$\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(+}\right)}}\right)x}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(+}\right)}x}\right)}\right)=x,$$

$$\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{+}}\right)x}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{+}x}\right)}\right)=x,\quad\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{+}x}\right)x^{+}}\right)=\mathopen{}\mathclose{{}\left(x^{+}\mathopen{}\mathclose{{}\left(xx^{+}}\right)}\right)=x^{+},$$

$$\displaystyle\mathfrak{s}:P\rightarrow\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\!P},\qquad x\mapsto\lambda{}_{x},$$

$$\displaystyle\mathfrak{\mathfrak{t}}:P\rightarrow\mathopen{}\mathclose{{}\left\{\rho}\right\}_{\!P},\qquad x\mapsto\rho{}_{x},$$

$$\mathopen{}\mathclose{{}\left(\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)\,x}\right)=x,\quad\mathopen{}\mathclose{{}\left(x\,\mathfrak{t}\mathopen{}\mathclose{{}\left(x}\right)}\right)=x.$$

$$x\mathopen{}\mathclose{{}\left(yx}\right)=\mathopen{}\mathclose{{}\left(xy}\right)z,\quad\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)\,x=x,\quad x\,\mathfrak{t}\mathopen{}\mathclose{{}\left(x}\right)=x.$$

$$\displaystyle\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}}\right)x}\right)=\mathopen{}\mathclose{{}\left(x\overline{x}x\overline{x}x}\right)=\mathopen{}\mathclose{{}\left(x\overline{x}x}\right)=x,$$

$$\displaystyle\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}}\right)x\mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}}\right)}\right)=\mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}x\overline{x}x\overline{x}}\right)=\mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}x\overline{x}}\right)=\mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}}\right),$$

$$\displaystyle\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right)\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right)\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right)}\right)=$$

$$\displaystyle\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ij}\right)^{-1}\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(ij}\right)^{-1}}\right)i}\right)=\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right),$$

$$\displaystyle\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right)\mathopen{}\mathclose{{}\left(ij}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(ij}\right)^{-1}\mathopen{}\mathclose{{}\left(ij}\right)}\right)=\mathopen{}\mathclose{{}\left(ij}\right),$$

$$\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(ji}\right)\mathopen{}\mathclose{{}\left(ij}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(ij}\right)}\right)=\mathopen{}\mathclose{{}\left(ij}\right),\quad\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ji}\right)\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(ji}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ji}\right)\mathopen{}\mathclose{{}\left(ji}\right)}\right)=\mathopen{}\mathclose{{}\left(ji}\right).$$

$$\mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(xx^{-1}xyy^{-1}y}\right)=\mathopen{}\mathclose{{}\left(xyy^{-1}x^{-1}xy}\right)$$

$$\mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right)=\mathopen{}\mathclose{{}\left(y^{-1}yy^{-1}x^{-1}xx^{-1}}\right)=\mathopen{}\mathclose{{}\left(y^{-1}x^{-1}xyy^{-1}x^{-1}}\right),$$

$$\mathfrak{i}:P\to P,x\mapsto x^{-1}$$

$$\displaystyle\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\,\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)}\right)\,x}\right)=x,\quad\mathopen{}\mathclose{{}\left(x\,\mathopen{}\mathclose{{}\left(\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)\,x}\right)}\right)=x,$$

$$\displaystyle\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)\,x}\right)\,\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)}\right)=\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right),\quad\mathopen{}\mathclose{{}\left(\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)\,\mathopen{}\mathclose{{}\left(x\,\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)}\right)}\right)=\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right).$$

Full text

1. Introduction

A binary operation $\mathfrak{m}$ on a set $S$ is usually defined as a mapping that assigns some \mathfrak{m}\mathopen{}\mathclose{{}\left(x,y}\right)\in S to every pair \mathopen{}\mathclose{{}\left(x,y}\right)\in S\times S. Algebraists have been somewhat reluctant to work with partial functions, in particular partial binary operations, where \mathfrak{m}\mathopen{}\mathclose{{}\left(x,y}\right) is defined only for all \mathopen{}\mathclose{{}\left(x,y}\right) in some subset of $S\times S$, often called the domain of definition of $\mathfrak{m}$. For example, in the early 1950s Wagner [17] pointed out that if an empty partial transformation is interpreted as the empty relation $\emptyset$ then composition of partial transformations can be regarded as being defined for every pair of transformations. Specifically, a partially defined binary system of non-empty partial transformations can be reduced to a semigroup which may contain an empty partial transformation by interpreting composition of partial transformations as composition of binary relations. This observation may have contributed to the acceptance of the notion of a binary system of partial transformations [8]. On the other hand, the reformulation alleviated the need to come to terms with partial binary operations as such, thus possibly leaving significant research questions unnoticed and unanswered.

One kind of reason why algebraists have hesitated to embrace partial operations has to do with the logical status of expressions such as \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)=y and \mathfrak{m}\mathopen{}\mathclose{{}\left(x,y}\right)=z when $\mathfrak{f}$ and $\mathfrak{m}$ are partial functions. As Burmeister [1] explains,

[a] first order language for algebraic systems is usually based on an appropriate notion of equations. Such a notion […] has already been around for quite a long time, but approaches as by Kleene […], Ebbinghaus […], Markwald […] and others (cf. also [Schein]) used then a three valued logic for the whole language (which might have deterred algebraists from using it). (pp. 306–7).

One might argue that if \mathopen{}\mathclose{{}\left(x,y}\right) does not belong to the domain of definition of $\mathfrak{m}$ then the assertion \mathfrak{m}\mathopen{}\mathclose{{}\left(x,y}\right)=z is meaningless, neither true nor false. This would seem to imply that we need some three-valued logic, where an assertion is not necessarily either true or false, similar to Kleene’s three-valued logic [10]. This approach is very problematic, however; one need only contemplate the meaning of an implication containing an assertion assumed to be neither true nor false to appreciate the complications that the use of a three-valued logic would entail.

Fortunately, two-valued logic suffices to handle partial functions, including partial binary operations. In particular, three-valued logic is not needed if expressions of the form \mathfrak{f}\mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{n}}\right)=y are used only if \mathopen{}\mathclose{{}\left(x_{1},\ldots,x_{n}}\right) belongs to the domain of definition of $\mathfrak{f}$. Burmeister [1] elaborated a formal logic with partial functions based on this idea. Another, simpler way to stay within two-valued logic when dealing with partial functions is sketched in Section 2 below.

But there is also another important kind of reason why algebraists have shunned partial binary operations. Ljapin and Evseev [13] note that

[it] often turns out that an idea embodying one clearly defined concept in the theory of total operations corresponds to several mutually inequivalent notions in the theory of partial operations, each one reflecting one or another aspect of the idea. (p. 17).

For example, in a group-like system with a partial binary operation, multiplication of elements can be associative in different ways, there are several types of identities and inverses, and different kinds of subsystems and homomorphisms can be distinguished. While this complicates algebraic theories using partial operations, it may be the case that these are differences that make a difference. Maybe the proliferation of notions just makes the theory richer and deeper, leading to a more profound understanding of the simpler special cases. Whether complexity equates richness and profoundness in this connection or not is a question that cannot really be answered a priori; the answer must be based on experience from more or less successful use of partial operations in applications such as those in this article.

In sum, there are no immediate reasons to avoid partial functions and operations. Particularly in view of the fact that some systems with partially defined operations, such as categories and groupoids, have received much attention for many years now, it would not be unreasonable to use a general theory of partial operations as a foundation for a general theory of total operations. This has not yet happened in mainstream mathematics, however: the mainstream definition of an algebra in Universal Algebra still uses total operations, not partially defined operations. (While partial operations are not ignored, they are typically treated as operations with special properties rather than as operations of the most general form.) This article is a modest attempt to fill a little of the resulting void by generalizing a theory employing total operations to a theory using partial operations. More concretely, we are concerned with “magmoids”: generalizations of magmas obtained by replacing the total binary operation by a partial operation. Specifically, “group-like” magmoids are considered; these generalize group-like magmas such as semigroups (without zeros), monoids and groups.

Section 2 contains the definitions of magmoids and other fundamental concepts, and introduces a convenient notation applicable to partial binary operations and other partial mappings. It is also shown that the basic concepts can be defined in a way that does not lead to any logical difficulties. Sections 3 and 4 deal with the many ways in which the notions of associative multiplication, identities and inverses from group theory can be generalized when the total binary operation is replaced by a partial binary operation; some other concepts related to identities and inverses are considered in Section 5. In Sections 6 and 7 a taxonomy of group-like magmoids based on the distinctions presented in Section 4 is developed. Some results that connect the group-like magmoids in Section 6 to those in Section 7 are proved in Section 8.

Much unconventional terminology is introduced in this article. This is because the core of the article is a systematic classification of some group-like magmoids, and the terminology reflects the logic of this classification. In some cases, the present terminology overlaps with traditional terminology, but the new terms are not meant to replace other, commonly used names of familiar concepts. Rather, the naming scheme used here is intended to call attention to similarities and differences between the notions distinguished.

2. Partial mappings and magmoids

2.1. Partial mappings

Let $X_{1},\ldots,X_{n}$ be non-empty sets. An n-ary relation on $X_{1},\ldots,X_{n}$, denoted $\mathfrak{r}:X_{1},\ldots,X_{n}$, or just $\mathfrak{r}$ when $X_{1},\ldots,X_{n}$ need not be specified, is a tuple

[TABLE]

where $\rho\subseteq X_{1}\!\times\!\ldots\!\times\!X_{n}$. The set $\rho$, also denoted $\gamma_{\mathfrak{r}}$, is the graph of $\mathfrak{r}$. The empty relation on $X_{1},\ldots,X_{n}$ is the tuple \mathopen{}\mathclose{{}\left(\emptyset,X_{1},\ldots,X_{n}}\right). The set

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a subset of $X_{i}$ denoted $\mathrm{pr}_{i}\,\rho$, is called the i:th projection of $\rho$. Note that if some projection of $\rho$ is the empty set then $\rho$ itself is the empty set.

A *binary (2-ary) relation $\mathfrak{r}:X,Y$ *is thus a tuple

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such that $\rho\subseteq X\times Y$. For $\mathfrak{r}:X,Y$ we have \mathrm{pr}_{1}\,\rho=\mathopen{}\mathclose{{}\left\{x\mid\mathopen{}\mathclose{{}\left(x,y}\right)\in\mathtt{\rho}}\right\} and \mathrm{pr}_{2}\,\rho=\mathopen{}\mathclose{{}\left\{y\mid\mathopen{}\mathclose{{}\left(x,y}\right)\in\rho}\right\}. We call $\mathrm{pr}_{1}\,\rho$ the effective domain of $\mathfrak{r}$, denoted $\mathrm{edom}\;\mathtt{\mathfrak{r}}$, and $\mathrm{pr}_{2}\,\rho$ the effective codomain of $\mathfrak{r}$, denoted $\mathrm{ecod}\;\mathfrak{r}$. We also call $X$ the total domain of $\mathtt{\mathfrak{r}}$, denoted $\mathrm{tdom}\;\mathtt{\mathfrak{r}}$, and $Y$ the total codomain of $\mathtt{\mathfrak{r}}$, denoted $\mathrm{tcod}\;\mathfrak{r}$. A total binary relation is a binary relation $\mathfrak{r}$ such that $\mathrm{edom}\;\mathtt{\mathfrak{r}}=\mathrm{tdom}\;\mathtt{\mathfrak{r}}$, while a cototal binary relation is a binary relation $\mathfrak{r}$ such that $\mathrm{ecod}\;\mathfrak{r}=\mathrm{tcod}\;\mathfrak{r}$.

Definition 2.1**.**

A functional relation, or (partial) mapping, $\mathfrak{f}:X\rightarrow Y$ is a binary relation

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such that for each $x\in\mathrm{pr}_{1}\,\phi$ there is exactly one $y\in\mathrm{pr}_{2}\,\phi$ such that \mathopen{}\mathclose{{}\left(x,y}\right)\in\phi. A self-mapping on $X$ is a mapping $\mathfrak{f}:X\rightarrow X$.

We let \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)=y express the fact that \mathopen{}\mathclose{{}\left(x,y}\right)\in\gamma_{\mathfrak{f}}. Consistent with this, for any $x\in\mathrm{edom}\;\mathfrak{f}$, \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right) denotes the unique element of $Y$ such that \mathopen{}\mathclose{{}\left(x,\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right)\in\gamma_{\mathfrak{f}}.

Let $\mathfrak{f}$ be a self-mapping on $X$. Then \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right) denotes some $x\in X$ if and only if $x\in\mathrm{edom}\>\mathfrak{f}$; \mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right) denotes some $x\in X$ if and only if x,\mathsf{\mathfrak{f}}\mathopen{}\mathclose{{}\left(x}\right)\in\mathrm{edom}\>\mathfrak{f}; etc. We describe such situations by saying that \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right), \mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right), etc. are defined.

We let \mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right), \mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right)}\right) etc. express the fact that \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right), \mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right) etc. is defined.111One could extend the scope of this notation, letting \mathopen{}\mathclose{{}\left(x}\right),\mathopen{}\mathclose{{}\left(x^{\prime}}\right),\ldots mean that $x,x^{\prime},\ldots$ belong to $X$, but in this article I will adhere to the more familiar, light-weight notation when dealing with ’naked’ variables, writing \mathopen{}\mathclose{{}\left(\phi\mathopen{}\mathclose{{}\left(x}\right)}\right)=y rather than \mathopen{}\mathclose{{}\left(\phi\mathopen{}\mathclose{{}\left(x}\right)}\right)=\mathopen{}\mathclose{{}\left(x^{\prime}}\right) or \mathopen{}\mathclose{{}\left(\phi\mathopen{}\mathclose{{}\left(x}\right)}\right)=\mathopen{}\mathclose{{}\left(y}\right), etc. We also use this notation embedded in expressions, letting \mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right)=y mean that \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right) is defined and \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)=y, letting \mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathfrak{g}\mathopen{}\mathclose{{}\left(x}\right)}\right) mean that \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right) and \mathfrak{g}\mathopen{}\mathclose{{}\left(x}\right) are defined and \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)=\mathfrak{g}\mathopen{}\mathclose{{}\left(x}\right), etc.

It is important to note that if \mathopen{}\mathclose{{}\left(x,y}\right)\in X\times Y but $x\notin\mathrm{edom}\>\mathfrak{f}=\mathrm{pr}_{1}\,\mathfrak{\gamma_{\mathfrak{f}}}$ then \mathopen{}\mathclose{{}\left(x,y}\right)\notin\gamma_{\mathfrak{f}}, so if \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right) is not defined then \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)=y is simply false, not meaningless. Also, \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)=\mathfrak{g}\mathopen{}\mathclose{{}\left(x}\right) is equivalent to the condition that there is some $y\in\mathrm{ecod}\,\mathfrak{f}\cap\mathrm{ecod}\,\mathfrak{g}$ such that \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)=y and \mathfrak{g}\mathopen{}\mathclose{{}\left(x}\right)=y, so such expressions do not present any new logical difficulties, although expressions such as \mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathfrak{g}\mathopen{}\mathclose{{}\left(x}\right)}\right) generally describe situations of more interest.

2.2. Binary operations and magmoids

Definition 2.2**.**

A (partial) binary operation on a non-empty set $X$ is a non-empty (partial) mapping

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A total binary operation on $X$ is a total mapping $\mathfrak{m}:X\times X\rightarrow X$.

A magmoid is a non-empty set $X$ equipped with a (partial) binary operation on $X$; a total magmoid, or magma, is a non-empty set $X$ equipped with a total binary operation on $X$.

Recall that in Definition 2.1, we identified a (partial) mapping $\mathfrak{f}:X\rightarrow Y$ with a binary relation \mathopen{}\mathclose{{}\left(\phi,X,Y}\right) such that \mathopen{}\mathclose{{}\left(x,y}\right),\mathopen{}\mathclose{{}\left(x,y^{\prime}}\right)\in\phi implies $y=y^{\prime}$. We can similarly identify a (partial) binary operation $\mathfrak{m}:X\times X\rightarrow X$ with a ternary relation

[TABLE]

such that \mathopen{}\mathclose{{}\left(x,y,z}\right),\mathopen{}\mathclose{{}\left(x,y,z^{\prime}}\right)\in\mu implies $z=z^{\prime}$, letting \mathfrak{m}\mathopen{}\mathclose{{}\left(x,y}\right)=z mean that \mathopen{}\mathclose{{}\left(x,y,z}\right)\in\mu. In this case, we have \mathrm{edom}\>\mathfrak{m}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(x,y}\right)\mid\mathopen{}\mathclose{{}\left(x,y,z}\right)\in\mu}\right\}, $\mathrm{tdom}\>\mathfrak{m}=X\times X$, \mathrm{ecod}\>\mathfrak{m}=\mathopen{}\mathclose{{}\left\{z\mid\mathopen{}\mathclose{{}\left(x,y,z}\right)\in\mu}\right\} and $\mathrm{tcod}\>\mathfrak{m}=X$.

The notion of being defined for expressions involving a self-mapping can be extended in a natural way to expressions involving a binary operation. We say that $xy$ is defined if and only if \mathopen{}\mathclose{{}\left(x,y}\right)\in\mathrm{edom}\>\mathfrak{m}; that \mathopen{}\mathclose{{}\left(xy}\right)z is defined if and only if \mathopen{}\mathclose{{}\left(x,y}\right),\mathopen{}\mathclose{{}\left(xy,z}\right)\in\mathrm{edom}\>\mathfrak{m}; that z\mathopen{}\mathclose{{}\left(xy}\right) is defined if and only if \mathopen{}\mathclose{{}\left(x,y}\right),\mathopen{}\mathclose{{}\left(z,xy}\right)\in\mathrm{edom}\>\mathfrak{m}; and so on. Thus, if \mathopen{}\mathclose{{}\left(xy}\right)z or z\mathopen{}\mathclose{{}\left(xy}\right) is defined then $xy$ is defined. In analogy with the notation \mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right), we let \mathopen{}\mathclose{{}\left(xy}\right) mean that $xy$ is defined, \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) mean that \mathopen{}\mathclose{{}\left(xy}\right)z is defined, \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) mean that x\mathopen{}\mathclose{{}\left(yz}\right) is defined, etc. Note that there is no conflict between the usual function of parentheses, namely to specify priority of operations, and their additional use here to show that a function is defined for a certain argument.

It is clear that \mathopen{}\mathclose{{}\left(x,y}\right)\notin\mathrm{edom}\>\mathfrak{m} implies \mathfrak{m}\mathopen{}\mathclose{{}\left(x,y}\right)\neq z for every $z\in X$, so in this case, too, we do not have to deal with logical anomalies.

Remark 2.1*.*

We have implicitly used lazy evaluation of conjunctions in this section. That is, the conjunction $p\wedge q$ is evaluated step-by-step according to the following algorithm:

[TABLE]

For example, if $x\notin\mathrm{edom}\>\mathfrak{f}$ then \mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right) is not defined; if $x\in\mathrm{edom}\>\mathfrak{f}$ but instead \mathsf{\mathfrak{f}}\mathopen{}\mathclose{{}\left(x}\right)\notin\mathrm{edom}\>\mathfrak{f} then \mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right) is also not defined; otherwise, \mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right) is defined. Thus, the question if \mathfrak{f}\mathopen{}\mathclose{{}\left(\mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right)}\right) is defined does not arise before we know if \mathfrak{f}\mathopen{}\mathclose{{}\left(x}\right) is defined.

Remark*.*

Burmeister [1] added some new primitives to standard logic to handle partially defined functions, and similarly the present approach ultimately requires a slight modification of standard logic, namely in the interpretation of conjunctions. However, it is important to note that the dynamic (lazy) evaluation interpretation of conjunctions is fully consistent with the static standard interpretation of conjunctions in terms of truth tables.

3. Conditions used in basic definitions

A group is a magma where multiplication is associative, and where there is an identity element and an inverse for every element. In this section, we distinguish components of these three properties of groups as they apply to magmoids.

3.1. Associativity

In a magma $M$, an associative binary operation is one that satisfies the condition x\mathopen{}\mathclose{{}\left(yz}\right)=\mathopen{}\mathclose{{}\left(xy}\right)z for all $x,y,z\in M$. If the magma is regarded as a magmoid $P$, we write this as

• (TA)

\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) for all $x,y,z\in P$.

In a magmoid we can in addition define conditions that generalize (TA):

• (A1)

If \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) for all $x,y,z\in P$.

• (A2)

If \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) for all $x,y,z\in P$.

• (A3)

If \mathopen{}\mathclose{{}\left(xy}\right) and \mathopen{}\mathclose{{}\left(yz}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) for all $x,y,z\in P$.

These elementary conditions concern different aspects of associativity, and can be used as building blocks for constructing more complex conditions; see Section 4.1.

In semigroups, we can omit parentheses, writing x\mathopen{}\mathclose{{}\left(yz}\right) and \mathopen{}\mathclose{{}\left(xy}\right)z as $xyz$ without ambiguity. In fact, it can be shown by induction using x\mathopen{}\mathclose{{}\left(yz}\right)=\mathopen{}\mathclose{{}\left(xy}\right)z that we can write $x_{1}\cdots x_{n}$ for any $n$ without ambiguity; this is the so-called law of general associativity. Similarly, in a magmoid where (A1) and (A2) hold we can write \mathopen{}\mathclose{{}\left(x_{1}\cdots x_{n}}\right) without ambiguity when all subproducts are defined, so we have a general associativity law in this case, too.

Specifically, let $\pi_{x_{1},\ldots,x_{n}}$ denote any parenthesized products of $x_{1},\ldots,x_{n}$, in this order, for example, \mathopen{}\mathclose{{}\left(x_{1}\mathopen{}\mathclose{{}\left(\cdots}\right)}\right) or \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\cdots}\right)x_{n}}\right). If (A1) and (A2) hold and $\pi_{x_{1},\ldots,x_{n}}^{\prime}$ is a parenthesized product of the same kind then it can be shown by induction that

[TABLE]

For example, if \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(zu}\right)}\right)}\right) then

[TABLE]

In other words, $\pi_{x_{1},\ldots,x_{n}}$ is uniquely determined by the sequence $x_{1},\ldots,x_{n}$, so we can write any $\pi_{x_{1},\ldots,x_{n}}$ as \mathopen{}\mathclose{{}\left(x_{1}\cdots x_{n}}\right). For example, we can write \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)\mathopen{}\mathclose{{}\left(zu}\right)}\right) as \mathopen{}\mathclose{{}\left(xyzu}\right) without loss of information.

(A3) thus implies that if \mathopen{}\mathclose{{}\left(xy}\right) and \mathopen{}\mathclose{{}\left(yz}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)=\mathopen{}\mathclose{{}\left(xyz}\right). By repeatedly applying (A1) and (A2) together with (A3), we can generalize (A3); for example, if \mathopen{}\mathclose{{}\left(xyz}\right) and \mathopen{}\mathclose{{}\left(zuv}\right) then \mathopen{}\mathclose{{}\left(xyzuv}\right). We can also generalize (A3) by applying it repeatedly; for example, if \mathopen{}\mathclose{{}\left(xy}\right), \mathopen{}\mathclose{{}\left(yz}\right) and \mathopen{}\mathclose{{}\left(zu}\right) then \mathopen{}\mathclose{{}\left(xyzu}\right). Combining these two ways of generalizing (A3), it becomes possible to make inferences such as ”\mathrm{if}\;\mathopen{}\mathclose{{}\left(x_{1}x_{2}}\right)\;\mathrm{and}\;\mathopen{}\mathclose{{}\left(x_{2}x_{3}x_{4}x_{5}}\right)\;\mathrm{and}\;\mathopen{}\mathclose{{}\left(x_{5}x_{6}x_{7}}\right)\mathrm{\ then}\;\mathopen{}\mathclose{{}\left(x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}}\right)”.

Note that we can retain certain redundant inner parentheses for emphasis. For example, if \mathopen{}\mathclose{{}\left(xx^{-1}x}\right)=x then we can write \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}x}\right)y}\right)=\mathopen{}\mathclose{{}\left(xy}\right) instead of \mathopen{}\mathclose{{}\left(xx^{-1}xy}\right)=\mathopen{}\mathclose{{}\left(xy}\right) to clarify how the equality is established.

3.2. Identities

In a magma $M$, an identity is an element $e\in M$ such that $ex=xe=x$ for all $x\in M$. More generally, a left (resp. right) identity is an element $e\in M$ such that $ex=x$ (resp. $xe=x$) for all $x\in M$. In a magmoid $P$, the conditions defining left and right identities take the following forms:

• (TU1)

If $x\in P$ then \mathopen{}\mathclose{{}\left(ex}\right)=x.

• (TU2)

If $x\in P$ then \mathopen{}\mathclose{{}\left(xe}\right)=x.

In a magmoid we can in addition define conditions generalizing (TU1) and (TU2):

• (GU1)

If $x\in P$ and \mathopen{}\mathclose{{}\left(ex}\right) then \mathopen{}\mathclose{{}\left(ex}\right)=x.

• (GU2)

If $x\in P$ and \mathopen{}\mathclose{{}\left(xe}\right) then \mathopen{}\mathclose{{}\left(xe}\right)=x.

• (LU1)

There is some $x\in P$ such that \mathopen{}\mathclose{{}\left(ex}\right)=x.

• (LU2)

There is some $x\in P$ such that \mathopen{}\mathclose{{}\left(xe}\right)=x.

In Section 4.2, we define different types of identities (units) in magmoids by means of combinations of these conditions.

3.3. Inverses

In a magma $M$, an inverse of $x\in M$ is an element $x^{-1}\in M$ such that $xx^{-1}=x^{-1}x=e$ for some identity $e\in M$, while a right (resp. left) inverse of $x\in M$ is an element $x^{-1}\in M$ such that $xx^{-1}=e$ (resp. $x^{-1}x=e$) for some identity $e\in M$. We can also define a* right (resp. left) semi-inverse* of $x\in M$ as an element $x^{-1}\in M$ such that $xx^{-1}=e$ (resp. $x^{-1}x=e$) for some left or right identity $e\in M$; these are the most fundamental notions. In a magmoid $P$, the conditions defining a left or right semi-inverse $x^{-1}$ take the following forms:

• (TI1)

If $x\in P$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=e for some $e$ satisfying (TU1).

• (TI2)

If $x\in P$ then \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=e for some $e$ satisfying (TU1).

• (TI3)

If $x\in P$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=e for some $e$ satisfying (TU2).

• (TI4)

If $x\in P$ then \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=e for some $e$ satisfying (TU2).

In addition, we can define conditions that generalize (TI1) – (TI4):

• (GI1)

If $x\in P$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=e for some $e$ satisfying (GU1).

• (GI2)

If $x\in P$ then \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=e for some $e$ satisfying (GU1).

• (GI3)

If $x\in P$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=e for some $e$ satisfying (GU2).

• (GI4)

If $x\in P$ then \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=e for some $e$ satisfying (GU2).

• (LI1)

If $x\in P$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=e for some $e$ satisfying (LU1).

• (LI2)

If $x\in P$ then \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=e for some $e$ satisfying (LU1).

• (LI3)

If $x\in P$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=e for some $e$ satisfying (LU2).

• (LI4)

If $x\in P$ then \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=e for some $e$ satisfying (LU2).

In Section 4.3, different types of inverses in magmoids will be defined in terms of conditions of this kind.

4. Basic definitions

Some of the possible combinations of elementary conditions in Section 3 will be used in this section to define different types of associatitvity, units (identities) and inverses. There are two main themes in this section: one-sided versus two-sided notions, and local versus global notions.

4.1. Magmoids according to types

of associativity

Definition 4.1**.**

Let $P$ be a magmoid, $x,y,z\in P$. Consider the following conditions:

• (S1)

If \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) or if \mathopen{}\mathclose{{}\left(xy}\right) and \mathopen{}\mathclose{{}\left(yz}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right).

• (S2)

If \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) or if \mathopen{}\mathclose{{}\left(xy}\right) and \mathopen{}\mathclose{{}\left(yz}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right).

• (S3)

If \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) or \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) or if \mathopen{}\mathclose{{}\left(xy}\right) and \mathopen{}\mathclose{{}\left(yz}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right).

A left semigroupoid is a magmoid satisfying (S1), a right semigroupoid is a magmoid satisfying (S2), and a (two-sided) semigroupoid is a magmoid satisfying (S3).

It is clear that if $P$ is a magma and at least one of the conditions (S1) – (S3) is satisfied then $P$ is a semigroup; conversely, if $P$ is a semigroup then $P$ is a magma where (S1) – (S3) are trivial implications so that they are all satisfied.

In view of the symmetry between left and right semigroupoids essentially only two cases will be considered below, namely left (or right) semigroupoids and (two-sided) semigroupoids. For later use, we note that a left semigroupoid is a magmoid such that for all $x,y,z\in P$

[TABLE]

while a semigroupoid is a left semigroupoid such that in addition to (4.1) we have

[TABLE]

4.2. Types of units in magmoids

Definition 4.2**.**

Let $P$ be a magmoid, $x\in P$.

1. (1)

A (global) left unit is some $\epsilon\in P$ such that if \mathopen{}\mathclose{{}\left(\epsilon x}\right) then \mathopen{}\mathclose{{}\left(\epsilon x}\right)=x, while a (global) right unit is some $\varepsilon\in P$ such that if \mathopen{}\mathclose{{}\left(x\varepsilon}\right) then \mathopen{}\mathclose{{}\left(x\varepsilon}\right)=x. 2. (2)

A (global) two-sided unit is some $e\in P$ which is a (global) left unit and a (global) right unit.

We denote the set of left units, right units, and two-sided units in $P$ by \mathopen{}\mathclose{{}\left\{\epsilon}\right\}_{\!P}, \mathopen{}\mathclose{{}\left\{\varepsilon}\right\}_{\!P}, and \mathopen{}\mathclose{{}\left\{e}\right\}_{\!P}, respectively.

Definition 4.3**.**

Let $P$ be a magmoid, $x\in P$.

1. (1)

A local left unit for $x$ is some $\lambda_{x}\in P$ such that \mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)=x; a local right unit for $x$ is some $\rho_{x}\in P$ such that x=\mathopen{}\mathclose{{}\left(x\rho_{x}}\right). 2. (2)

A* twisted left unit $\varphi_{x}\in P$ for* $x$ is a local left unit for $x$ which is a right unit, while a* twisted right unit $\psi_{x}\in P$ for* $x$ is a local right unit for $x$ which is a left unit. 3. (3)

A *left effective unit *$\ell_{x}\in P$ for $x$ is a local left unit for $x$ which is a two-sided unit, while a *right effective unit *$r_{x}\in P$ for $x$ is a local right unit for $x$ which is a two-sided unit.

We denote the set of local left (resp. local right) units for $x\in P$ by \mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x} (resp. \mathopen{}\mathclose{{}\left\{\rho}\right\}_{x}), the set of twisted left (resp. twisted right) units for $x\in P$ by \mathopen{}\mathclose{{}\left\{\varphi}\right\}_{x} (resp. \mathopen{}\mathclose{{}\left\{\psi}\right\}_{x}), and the set of left effective (resp. right effective) units for $x\in P$ by \mathopen{}\mathclose{{}\left\{\ell}\right\}_{x} (resp. \mathopen{}\mathclose{{}\left\{r}\right\}_{x}). We also set \mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x}, \mathopen{}\mathclose{{}\left\{\rho}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x}, \mathopen{}\mathclose{{}\left\{\varphi}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{\varphi}\right\}_{x}, \mathopen{}\mathclose{{}\left\{\psi}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{\psi}\right\}_{x}, \mathopen{}\mathclose{{}\left\{\ell}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{\ell}\right\}_{x} and \mathopen{}\mathclose{{}\left\{r}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{r}\right\}_{x}.

4.3. Types of inverses in magmoids

Definition 4.4**.**

Let $P$ be a magmoid, $x\in P$.

1. (1)

A pseudoinverse of $x$ is some x^{\mathopen{}\mathclose{{}\left(-1}\right)}\in P such that \mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(-1}\right)}}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x} and \mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(-1}\right)}x}\right)\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x}. 2. (2)

A right preinverse of $x$ is some $x^{-1}\in P$ such that \mathopen{}\mathclose{{}\left(xx^{-1}}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x} and \mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x^{-1}}. 3. (3)

A left preinverse of $x$ is some $x^{-1}\in P$ such that \mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{\rho}\right\}{}_{x} and \mathopen{}\mathclose{{}\left(xx^{-1}}\right)\in\mathopen{}\mathclose{{}\left\{\rho}\right\}{}_{x^{-1}}. 4. (4)

A (two-sided) preinverse of $x$ is some $x^{-1}\in P$ such that we have \mathopen{}\mathclose{{}\left(xx^{-1}}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x}\cap\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x^{-1}} and \mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x}\cap\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x^{-1}}.

Thus, x^{\mathopen{}\mathclose{{}\left(-1}\right)} is a pseudoinverse of $x$ if and only if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(-1}\right)}}\right)x}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(-1}\right)}x}\right)}\right)=x, $x^{-1}$ is a right preinverse of $x$ if and only if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)=x and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)x^{-1}}\right)=x^{-1}, $x^{-1}$ is a left preinverse of $x$ if and only if \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=x and \mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=x^{-1}, and $x^{-1}$ is a preinverse of $x$ if and only if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=x and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)x^{-1}}\right)=\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=x^{-1}.

Let $\mathbf{J}$, $\mathbf{I}^{+}$, $\mathfrak{\mathbf{I}}^{*}$ and $\mathfrak{\mathbf{I}}$ be binary relations on a magmoid $P$ such that $x\,\mathbf{J}\,\overline{x}$ if and only if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\overline{x}}\right)x}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\overline{x}x}\right)}\right)=x, $x\,\mathbf{I}^{+}\,\overline{x}$ if and only if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\overline{x}}\right)x}\right)=x and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\overline{x}x}\right)\overline{x}}\right)=\overline{x}, $x\,\mathbf{I}^{*}\,\overline{x}$ if and only if \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\overline{x}x}\right)}\right)=x and \mathopen{}\mathclose{{}\left(\overline{x}\mathopen{}\mathclose{{}\left(x\overline{x}}\right)}\right)=\overline{x}, and $x\,\mathbf{I}\,\overline{x}$ if and only if $x\,\mathfrak{\mathbf{I}^{+}}\,\overline{x}$ and $x\,\mathbf{I}^{*}\,\overline{x}$. In terms of these relations, $\overline{x}$ is a pseudoinverse of $x$ if and only if $x\,\mathbf{J}\,\overline{x}$, a right preinverse of $x$ if and only if $x\,\mathbf{I}^{+}\,\overline{x}$, a left preinverse of $x$ if and only if $x\,\mathbf{I}^{*}\,\overline{x}$, and a preinverse of $x$ if and only if $x\,\mathbf{I}\,\overline{x}$. Note that $\mathbf{I}^{+}$, $\mathbf{I}^{*}$ and $\mathbf{I}$ are symmetric relations.

It is useful to have some special notation for sets of pseudoinverses and preinverses, and we set \mathbf{J}\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\mathbf{J}\,\overline{x}}\right\}, \mathbf{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\mathbf{I}^{+}\,\overline{x}}\right\}, \mathbf{I}^{*}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\mathbf{I}^{*}\,\overline{x}}\right\} and \mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\mathbf{I}\,\overline{x}}\right\}.

Definition 4.5**.**

Let $P$ be a magmoid, $x\in P$.

1. (1)

A strong pseudoinverse of $x$ is a pseudoinverse x^{\mathopen{}\mathclose{{}\left(-1}\right)} of $x$ such that \mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(-1}\right)}}\right),\mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(-1}\right)}x}\right)\in\mathopen{}\mathclose{{}\left\{e}\right\}{}_{P}. 2. (2)

A strong right preinverse of $x$ is a right preinverse $x^{-1}$ of $x$ such that \mathopen{}\mathclose{{}\left(xx^{-1}}\right),\mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{\varepsilon}\right\}{}_{P}. 3. (3)

A strong left preinverse of $x$ is a left preinverse $x^{-1}$ of $x$ such that \mathopen{}\mathclose{{}\left(x^{-1}x}\right),\mathopen{}\mathclose{{}\left(xx^{-1}}\right)\in\mathopen{}\mathclose{{}\left\{\epsilon}\right\}{}_{P}. 4. (4)

A strong (two-sided) preinverse of $x$ is a preinverse $x^{-1}$ of $x$ such that \mathopen{}\mathclose{{}\left(xx^{-1}}\right),\mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{e}\right\}{}_{P}.

By this definition, a strong right preinverse of $x$ is a right preinverse $x^{-1}$ of $x$ such that \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=\mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=y for all $y\in P$, and a strong preinverse of $x$ is a preinverse $x^{-1}$ of $x$ such that \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)\!=\!\mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)\!=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)y}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)y}\right)=y for all $y\in P$. Strong left preinverses and strong pseudoinverses have corresponding properties.

We let $x\,\boldsymbol{J}\,\overline{x}$, $x\,\boldsymbol{I}^{+}\,\overline{x}$, $x\,\boldsymbol{I}^{*}\,\overline{x}$ and $x\,\boldsymbol{I}\,\overline{x}$ mean that $\overline{x}$ is a strong pseudoinverse, strong right preinverse, strong left preinverse and strong preinverse of $x$, respectively. It is clear that $\boldsymbol{I}^{+}$, $\boldsymbol{I}^{*}$ and $\boldsymbol{I}$ are symmetrical relations. We also set \boldsymbol{J}\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\boldsymbol{J}\,\overline{x}}\right\}, \boldsymbol{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\boldsymbol{I}^{+}\,\overline{x}}\right\}, \boldsymbol{I}^{*}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\boldsymbol{I}^{*}\,\overline{x}}\right\} and \boldsymbol{I}\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\overline{x}\mid x\,\boldsymbol{I}\,\overline{x}}\right\}.

4.4. Canonical local units

Recall that if $x^{-1}$ is a left, right or two-sided preinverse of $x$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x}\cup\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x^{-1}} and \mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x}\cup\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x^{-1}}. We set \boldsymbol{\uplambda}_{x}=\boldsymbol{\uprho}_{x^{-1}}=\mathopen{}\mathclose{{}\left(xx^{-1}}\right), \boldsymbol{\uprho}_{x}=\boldsymbol{\uplambda}_{x^{-1}}=\mathopen{}\mathclose{{}\left(x^{-1}x}\right), and call such local units canonical local units. We may also use the notation \mathopen{}\mathclose{{}\left\{\boldsymbol{\uplambda}}\right\}_{x} (resp. \mathopen{}\mathclose{{}\left\{\boldsymbol{\uprho}}\right\}_{x}) for the set of canonical local left (resp. right) units for $x$, and the notation \mathopen{}\mathclose{{}\left\{\boldsymbol{\uplambda}}\right\}_{x^{-1}} (resp. \mathopen{}\mathclose{{}\left\{\boldsymbol{\uprho}}\right\}_{x^{-1}}) for the set of canonical local left (resp. right) units for $x^{-1}$. We also set \mathopen{}\mathclose{{}\left\{\boldsymbol{\uplambda}}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{\boldsymbol{\uplambda}}\right\}_{x} and \mathopen{}\mathclose{{}\left\{\boldsymbol{\uprho}}\right\}_{\!P}=\cup_{x\in P}\mathopen{}\mathclose{{}\left\{\boldsymbol{\uprho}}\right\}_{x}.

A canonical local unit of the form \mathopen{}\mathclose{{}\left(xx^{.-1}}\right) (resp. \mathopen{}\mathclose{{}\left(x^{-1}x}\right)) such that \mathopen{}\mathclose{{}\left(xx^{\prime}}\right)=\mathopen{}\mathclose{{}\left(xx^{\prime\prime}}\right) (resp. \mathopen{}\mathclose{{}\left(x^{\prime}x}\right)=\mathopen{}\mathclose{{}\left(x^{\prime\prime}x}\right)) for any inverses $x^{\prime},x^{\prime\prime}$ of $x$ is said to be unique.

If we regard $x^{-1}$ not as a preinverse of $x\in P$ but just as an element $x^{-1}\in P$, we write \boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)} instead of $\boldsymbol{\uplambda}_{x^{-1}}$ and \boldsymbol{\uprho}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)} instead of $\boldsymbol{\uprho}_{x^{-1}}$, setting

[TABLE]

for some preinverse \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1} of $x^{-1}$. Note, though, that if $x\,\mathbf{I}\,x^{-1}$ then $x^{-1}\,\mathbf{I}\,x$, so x\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{x^{-1}}, so if \boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)} is unique then \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}}\right), meaning that \boldsymbol{\uplambda}_{x^{-1}}=\boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)}; similarly, if \boldsymbol{\uprho}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)} is unique then \boldsymbol{\uprho}_{x^{-1}}=\boldsymbol{\uprho}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)}. These identities hold for left and right preinverses as well, since $\mathbf{I}^{+}$ and $\mathbf{I}^{*}$ are symmetric relations.

Remark 4.1*.*

It is natural to write $y=\lambda_{x}$ when \mathopen{}\mathclose{{}\left\{y}\right\}=\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x}, $y=\rho_{x}$ when \mathopen{}\mathclose{{}\left\{y}\right\}=\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x}, $y=x^{-1}$ when \mathopen{}\mathclose{{}\left\{y}\right\}=\mathbf{I}{}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}, \mathopen{}\mathclose{{}\left\{y}\right\}=\mathbf{I}^{*}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}, \mathopen{}\mathclose{{}\left\{y}\right\}=\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{x}, \mathopen{}\mathclose{{}\left\{y}\right\}=\boldsymbol{I}{}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}, \mathopen{}\mathclose{{}\left\{y}\right\}=\boldsymbol{I}^{*}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x} or \mathopen{}\mathclose{{}\left\{y}\right\}=\boldsymbol{\boldsymbol{I}}\mathopen{}\mathclose{{}\left\{}\right\}_{x}, and so on.

5. Idempotents and involution

5.1. Idempotents in magmoids

Definition 5.1**.**

An* idempotent* in a magmoid $P$ is some $i\in P$ such that \mathopen{}\mathclose{{}\left(ii}\right)=i.

Proposition 5.1**.**

Let $P$ be a magmoid, $i\in P$. If \mathopen{}\mathclose{{}\left(ii}\right)=i then i\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{i}\cap\mathopen{}\mathclose{{}\left\{\rho}\right\}_{i}.

Proof.

If \mathopen{}\mathclose{{}\left(ii}\right)=i then i\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{i} and i\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{i}. ∎

Corollary 5.1**.**

Let $P$ be a magmoid with unique local units, $i\in P$. If \mathopen{}\mathclose{{}\left(ii}\right)=i then $i=\lambda_{i}=\rho_{i}$.

Proposition 5.2**.**

Let $P$ be a magmoid, $i\in P$. If \mathopen{}\mathclose{{}\left(ii}\right)=i then i\in\mathbf{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{i}, i\in\mathbf{I}^{*}\!\mathopen{}\mathclose{{}\left\{}\right\}_{i} and i\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{i}.

Proof.

If \mathopen{}\mathclose{{}\left(ii}\right)=i then \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ii}\right)i}\right)=\mathopen{}\mathclose{{}\left(i\mathopen{}\mathclose{{}\left(ii}\right)}\right)=i, so $i\,\mathfrak{\mathbf{I}^{+}}i$, $i\,\mathbf{I}^{*}i$ and $i\,\mathbf{I}\,i$. ∎

Corollary 5.2**.**

Let $P$ be a magmoid where right preinverses, left preinverses or (two-sided) preinverses are unique, $i\in P$. If \mathopen{}\mathclose{{}\left(ii}\right)=i then $i^{-1}=i$.

Proof.

We have i\in\mathbf{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{i}=\mathopen{}\mathclose{{}\left\{i^{-1}}\right\} or i\in\mathbf{I}^{*}\!\mathopen{}\mathclose{{}\left\{}\right\}_{i}=\mathopen{}\mathclose{{}\left\{i^{-1}}\right\} or i\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{i}=\mathopen{}\mathclose{{}\left\{i^{-1}}\right\}. ∎

Proposition 5.3**.**

Let $P$ be a left (resp. right) semigroupoid with unique right (resp. left) preinverses, or a semigroupoid with unique preinverses. Then \mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \mathopen{}\mathclose{{}\left(x^{-1}x}\right) are idempotents for every $x\in P$, and for every idempotent $i\in P$ there is some $x\in P$ such that i=\mathopen{}\mathclose{{}\left(xx^{-1}}\right)=\mathopen{}\mathclose{{}\left(x^{-1}x}\right).

Proof.

It suffices two consider the first two cases. In both of these, \mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \mathopen{}\mathclose{{}\left(x^{-1}x}\right) so that \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)}\right), \mathopen{}\mathclose{{}\left(x^{-1}\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)}\right), \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)x^{-1}}\right)x}\right) and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)x^{-1}}\right), and in a left (resp. right) semigroupoid we have

[TABLE]

Conversely, if $i\in P$ is an idempotent then $i=i^{-1}$ by Corollary 5.2, so i=\mathopen{}\mathclose{{}\left(ii}\right)=\mathopen{}\mathclose{{}\left(ii^{-1}}\right)=\mathopen{}\mathclose{{}\left(i^{-1}i}\right). ∎

5.2. Involution magmoids

A magma $M$ may be equipped with a total self-mapping $*:x\mapsto x^{*}$ such that \mathopen{}\mathclose{{}\left(x^{*}}\right)^{*}=x and \mathopen{}\mathclose{{}\left(xy}\right)^{*}=y^{*}x^{*}. A total self-mapping with these properties is an anti-endomorphism on $M$ by the second condition, and a bijection by the first condition,222If $x_{1}^{*}=x_{2}^{*}$ then x_{1}=\mathopen{}\mathclose{{}\left(x_{1}^{*}}\right)^{*}=\mathopen{}\mathclose{{}\left(x_{2}^{*}}\right)^{*}=x_{2}, and if $x\in M$ then x=\mathopen{}\mathclose{{}\left(x^{*}}\right)^{*}\in\mathopen{}\mathclose{{}\left(M^{*}}\right)^{*}\subseteq M^{*} since $M^{*}\subseteq M$, so $M\subseteq M^{*}$, so $M=M^{*}$. Hence, $x\mapsto x^{*}$ is injective and surjective. so $*$ is an anti-automorphism, called an involution for $M$.333This is the definition in semigroup theory; in general mathematics an involution is usually defined as a self-mapping $*$ such that \mathopen{}\mathclose{{}\left(x^{*}}\right)^{*}=x. This notion can be generalized to magmoids.

Definition 5.2**.**

A (total) involution magmoid is a magmoid $P$ equipped with a total mapping

[TABLE]

such that \mathopen{}\mathclose{{}\left(x^{*}}\right)^{*}=x and if \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)^{*}=\mathopen{}\mathclose{{}\left(y^{*}x^{*}}\right) for all $x,y\in P$. We call the function $x\mapsto x^{*}$ an involution and $x^{*}$ the involute of $x$.

Involutes are inverse-like elements, but while inverses are defined in terms of units of various kinds (as elaborated in Sections 3.3 and 4.3), involutes are defined without reference to units, so involutes generalize inverses to situations where units may not be available. Conversely, however, unit-like elements may be defined in terms of involutes.

Definition 5.3**.**

A unity in a magmoid $P$ with involution $*$ is some $u\in P$ such that $u^{*}=u$.

For any $x\in P$, \mathopen{}\mathclose{{}\left(xx^{*}}\right) and \mathopen{}\mathclose{{}\left(x^{*}x}\right) are unities since \mathopen{}\mathclose{{}\left(xx^{*}}\right)^{*}=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{*}}\right)^{*}x^{*}}\right)=\mathopen{}\mathclose{{}\left(xx^{*}}\right) and \mathopen{}\mathclose{{}\left(x^{*}x}\right)^{*}=\mathopen{}\mathclose{{}\left(x^{*}\mathopen{}\mathclose{{}\left(x^{*}}\right)^{*}}\right)=\mathopen{}\mathclose{{}\left(x^{*}x}\right). If $P$ is a semigroupoid and \mathopen{}\mathclose{{}\left(xx^{*}x}\right)=x then \mathopen{}\mathclose{{}\left(x^{*}xx^{*}x}\right)=\mathopen{}\mathclose{{}\left(x^{*}x}\right) and \mathopen{}\mathclose{{}\left(xx^{*}xx^{*}}\right)=\mathopen{}\mathclose{{}\left(xx^{*}}\right), so that \mathopen{}\mathclose{{}\left(xx^{*}}\right) and \mathopen{}\mathclose{{}\left(x^{*}x}\right) are idempotents.

We can use unities to define a kind of inverses, just as we used units to define inverses in Section 4.3. Let $P$ be an involution magmoid, $x\in P$. An involution pseudoinverse of $x$ is some x^{\mathopen{}\mathclose{{}\left(+}\right)}\in P such that

[TABLE]

where \mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(+}\right)}}\right) and \mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(+}\right)}x}\right) are unities, while an involution preinverse of $x$ is some $x^{+}\in P$ such that

[TABLE]

where \mathopen{}\mathclose{{}\left(xx^{+}}\right) and \mathopen{}\mathclose{{}\left(x^{+}x}\right) are unities. It is easy to show that in a semigroupoid there is at most one involution preinverse for each $x\in P$.

In the semigroupoid of all matrices over $\mathbb{R}$ or $\mathbb{C}$, where \mathopen{}\mathclose{{}\left(MN}\right) if and only if $M$ is a $p\times q$ matrix and $N$ is a $q\times r$ matrix, the transpose $A^{\mathrm{T}}$ and conjugate transpose $A^{\dagger}$ of a $p\times q$ matrix $A$ are involutes of $A$.444If $\alpha_{ji}=a_{ij}$, $\beta_{kj}=b_{jk}$ and $\gamma_{ki}=\sum_{j}a_{ij}b_{jk}$ then $\gamma_{ki}=\sum_{j}\beta{}_{kj}\alpha_{ji}$ and $\overline{\gamma_{ki}}=\sum_{j}\overline{\beta_{kj}}\overline{\alpha{}_{ji}}$. The unities are then symmetric or Hermitian matrices, that is, matrices such that $A^{\mathrm{T}}=A$ or $A^{\dagger}=A$. The involution preinverse of $A$ is the unique555Involution inverses are unique when they exist, and it can be shown that every matrix $A$ has an involution inverse $A^{+}$ with respect to the involutions $A\mapsto A^{\mathrm{T}}$ and $A\mapsto A^{\dagger}$. $q\times p$ matrix $A^{+}$ such that \mathopen{}\mathclose{{}\left(AA^{+}A}\right)=A, \mathopen{}\mathclose{{}\left(A^{+}AA^{+}}\right)=A^{+} and such that the $p\times p$ matrix \mathopen{}\mathclose{{}\left(AA^{+}}\right) and the $q\times q$ matrix \mathopen{}\mathclose{{}\left(A^{+}A}\right) are unities, that is, symmetric or Hermitian matrices for which \mathopen{}\mathclose{{}\left(AA^{+}}\right)^{*}=\mathopen{}\mathclose{{}\left(AA^{+}}\right) and \mathopen{}\mathclose{{}\left(A^{+}A}\right)^{*}=\mathopen{}\mathclose{{}\left(A^{+}A}\right). $A^{+}$ is known as the Moore-Penrose inverse of $A$.

Total involutions can be generalized to partial involutions; this notion is also of interest. For example, it is easy to verify that the partial function $A\mapsto A^{-1}$, which associates every invertible matrix with its inverse, is a partial involution in the semigroupoid of matrices over $\mathbb{R}$ or $\mathbb{C}$. In every subsemigroupoid of invertible $n\times n$ matrices over $\mathbb{R}$ or $\mathbb{C}$, $A\mapsto A^{-1}$ is a total involution, and $A^{-1}$ is an involution preinverse as well as an involute, so $A^{+}=A^{-1}$ since involution preinverses are unique – recall that the Moore-Penrose inverse generalizes the ordinary matrix inverse.

The relationship between involution magmoids and corresponding semiheapoids is briefly described in Appendix A.

6. Prepoloids and related magmoids

The magmoids considered in this section are equipped with local units. In the literature, the focus is on the special case when these magmoids are magmas, namely semigroups, but here we generalize such magmas to magmoids.

6.1. The prepoloid family

Definition 6.1**.**

Let $P$ be a semigroupoid, Then $P$ is

1. (1)

a* prepoloid666’Prepoloids’ generalize the ’poloids’ considered in Section 7. The term ’poloid’ for a generalized monoid was introduced and motivated in [9]. when there is a local left unit $\lambda_{x}\in P$ and a local right unit $\rho_{x}\in P$ for every $x\in P$;* 2. (2)

a pregroupoid777Kock [11] lets the term ’pregroupoid’ refer to a set with a partially defined ternary operation. when $P$ is a prepoloid such that for every $x\in P$ there is a preinverse $x^{-1}\in P$ of $x$.

This means that a prepoloid $P$ is a semigroupoid such that for every $x\in P$ there are local units $\lambda{}_{x},\rho{}_{x}$ such that \mathopen{}\mathclose{{}\left(\lambda{}_{x}x}\right)=\mathopen{}\mathclose{{}\left(x\rho{}_{x}}\right)=x. A pregroupoid is a prepoloid $P$ such that for every $x\in P$ there is some $x^{-1}\in P$ such that \mathopen{}\mathclose{{}\left(xx^{-1}}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x}\cap\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x^{-1}} and \mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x}\cap\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x^{-1}}.

Prepoloids

Proposition 6.1**.**

Let $P$ be a prepoloid, $x,y\in P$, \rho_{x}\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x} and \lambda_{y}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{y}. If $\rho_{x}=\lambda_{y}$ then \mathopen{}\mathclose{{}\left(xy}\right).

Proof.

If $\rho_{x}=\lambda_{y}$ then \mathopen{}\mathclose{{}\left(\rho_{x}y}\right) since \mathopen{}\mathclose{{}\left(\lambda_{y}y}\right), so \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\rho_{x}}\right)y}\right)=\mathopen{}\mathclose{{}\left(xy}\right) since \mathopen{}\mathclose{{}\left(x\rho_{x}}\right)=x. ∎

Proposition 6.2**.**

Let $P$ be a prepoloid with unique local units, $x\in P$. Then \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)=\lambda_{x}=\lambda_{\lambda_{x}}=\rho{}_{\lambda_{x}} and \mathopen{}\mathclose{{}\left(\rho_{x}\rho_{x}}\right)=\rho_{x}=\rho_{\rho_{x}}=\lambda{}_{\rho_{x}}.

Proof.

If $x\in P$ then x=\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)=\mathopen{}\mathclose{{}\left(\lambda_{x}\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)x}\right), so \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\lambda_{x}}\right\}. Also, \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)=\lambda_{x} implies that \lambda_{x}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\lambda_{x}}=\mathopen{}\mathclose{{}\left\{\lambda_{\lambda_{x}}}\right\} and \lambda_{x}\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{\lambda_{x}}=\mathopen{}\mathclose{{}\left\{\rho{}_{\lambda_{x}}}\right\}. It is proved similarly that \mathopen{}\mathclose{{}\left(\rho_{x}\rho_{x}}\right)=\rho_{x}=\rho_{\rho_{x}}=\lambda{}_{\rho_{x}}. ∎

By Propositions 5.1 and 6.2, an element of a prepoloid with unique local units is thus a local unit if and only if it is an idempotent.

Proposition 6.3**.**

Let $P$ be a prepoloid with unique local units, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \lambda_{\mathopen{}\mathclose{{}\left(xy}\right)}=\lambda_{x} and \rho_{\mathopen{}\mathclose{{}\left(xy}\right)}=\rho_{y}.

Proof.

If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)y}\right)=\mathopen{}\mathclose{{}\left(\lambda_{x}\mathopen{}\mathclose{{}\left(xy}\right)}\right), so \lambda_{x}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{\lambda_{\mathopen{}\mathclose{{}\left(xy}\right)}}\right\}. Similarly, \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(y\rho_{y}}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)\rho_{y}}\right), so \rho_{y}\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{\rho_{\mathopen{}\mathclose{{}\left(xy}\right)}}\right\}. ∎

A prepoloid $P$ with unique local units can be equipped with unique surjective functions

[TABLE]

such that

[TABLE]

By Proposition 6.2, \mathfrak{s}\mathopen{}\mathclose{{}\left(\lambda_{x}}\right)=\lambda_{x} for all \lambda_{x}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\!P} and \mathfrak{\mathfrak{t}}\mathopen{}\mathclose{{}\left(\rho_{x}}\right)=\rho_{x} for all \rho_{x}\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{\!P}. A prepoloid with unique local units can thus be regarded as a semigroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m}}\right) expanded to a prepoloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s},\mathfrak{t}}\right) characterized by the uniqueness property and the identities (4.1), (4.2), and (6.1).

Note that there may exist more than one function \mathfrak{s}:P\rightarrow\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\!P} such that \mathopen{}\mathclose{{}\left(\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)x}\right)=x for all $x\in P$, and more than one function \mathfrak{t}:P\rightarrow\mathopen{}\mathclose{{}\left\{\rho}\right\}_{\!P} such that \mathopen{}\mathclose{{}\left(x\mathfrak{t}\mathopen{}\mathclose{{}\left(x}\right)}\right)=x for all $x\in P$ (see Example 6.1).

We call a prepoloid which admits not necessarily unique functions $\mathfrak{s},\mathfrak{\mathfrak{t}}:P\rightarrow P$ satisfying (6.1) a bi-unital prepoloid; those bi-unital prepoloids which are magmas are bi-unital semigroups. These are thus characterized by the identities

[TABLE]

The class of bi-unital semigroups includes many types of semigroups studied in the literature such as the function systems of Schweitzer and Sklar [15], abundant semigroups [4], adequate semigroups [3], Ehresmann semigroups [12], ample semigroups and restriction semigroups (see, e.g., [7]), and regular and inverse semigroups with $\mathfrak{s}$ and $\mathfrak{t}$ defined by \mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)=\mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \mathfrak{t}\mathopen{}\mathclose{{}\left(x}\right)=\mathopen{}\mathclose{{}\left(x^{-1}x}\right).

As we have seen, several identities, such as \mathfrak{s}\mathopen{}\mathclose{{}\left(\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)}\right)=\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right), \mathopen{}\mathclose{{}\left(\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)}\right)=\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right) and \mathfrak{s}\mathopen{}\mathclose{{}\left(xy}\right)=\mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right), can be derived from the assumption that local units are unique. Bi-unital pregroupoids and semigroups where this is not postulated can be required to satisfy other conditions in order to have desirable properties; various such requirements are used to characterize the bi-unital semigroups found in the literature.

Pregroupoids (1)

It is a well-known result in semigroup theory that a semigroup $S$ has a pseudoinverse x^{\mathopen{}\mathclose{{}\left(-1}\right)} such that xx^{\mathopen{}\mathclose{{}\left(-1}\right)}x=x for every $x\in S$ if and only if $S$ has a so-called generalized inverse $x^{-1}$ such that $xx^{-1}x=x$ and $x^{-1}xx^{-1}=x^{-1}$ for every $x\in S$. A regular semigroup can thus be defined by either condition. This result can be generalized to semigroupoids.

Proposition 6.4**.**

Let $P$ be a semigroupoid. Each $x\in P$ has a preinverse $x^{-1}\in P$ if and only if each $x\in P$ has a pseudoinverse x^{\mathopen{}\mathclose{{}\left(-1}\right)}\in P.

Proof.

Trivially, each preinverse $x^{-1}$ of $x$ is a pseudoinverse x^{\mathopen{}\mathclose{{}\left(-1}\right)} of $x$. Conversely, if $\overline{x}$ is a pseudoinverse of $x$ then \mathopen{}\mathclose{{}\left(x\overline{x}x}\right), \mathopen{}\mathclose{{}\left(x\overline{x}}\right) and \mathopen{}\mathclose{{}\left(\overline{x}x}\right). Thus, \mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}}\right), \mathopen{}\mathclose{{}\left(x\overline{x}x\overline{x}x}\right) and \mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}x\overline{x}x\overline{x}}\right), and as \mathopen{}\mathclose{{}\left(x\overline{x}x}\right)=x we have

[TABLE]

so x\,\mathbf{I}\,\mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}}\right), meaning that \mathopen{}\mathclose{{}\left(\overline{x}x\overline{x}}\right)\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{x}. ∎

Recall that an inverse semigroup can be defined as a regular semigroup whose preinverses are unique, or as a regular semigroup whose idempotents commute. These two characterizations are equivalent also when inverse semigroups are generalized to pregroupoids.

Lemma 6.1**.**

Let $P$ be a pregroupoid with unique preinverses, $i,j\in P$. If $i,j$ are idempotents and \mathopen{}\mathclose{{}\left(ij}\right) then \mathopen{}\mathclose{{}\left(ij}\right) and \mathopen{}\mathclose{{}\left(ji}\right) are idempotents.

Proof.

We have \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(ij}\right)^{-1}\mathopen{}\mathclose{{}\left(ij}\right)}\right), so \mathopen{}\mathclose{{}\left(ij\mathopen{}\mathclose{{}\left(ij}\right)^{-1}ij}\right), so \mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right). Since \mathopen{}\mathclose{{}\left(ii}\right), \mathopen{}\mathclose{{}\left(jj}\right) and \mathopen{}\mathclose{{}\left(ij}\right), \mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right) implies \mathopen{}\mathclose{{}\left(ijj\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right) and \mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}iij}\right). Thus, \mathopen{}\mathclose{{}\left(ijj\mathopen{}\mathclose{{}\left(ij}\right)^{-1}iij}\right) and \mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}iijj\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right), and we have

[TABLE]

so \mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right) is an idempotent and \mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right)\,\mathfrak{\mathbf{I}}\,\mathopen{}\mathclose{{}\left(ij}\right). Using the fact that preinverses are unique so that idempotents are preinverses of themselves by Corollary 5.2, we conclude that \mathopen{}\mathclose{{}\left(ij}\right) is an idempotent since \mathopen{}\mathclose{{}\left(ij}\right)=\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right)^{-1}=\mathopen{}\mathclose{{}\left(j\mathopen{}\mathclose{{}\left(ij}\right)^{-1}i}\right).

Finally, if \mathopen{}\mathclose{{}\left(ij}\right) is an idempotent then \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ij}\right)\mathopen{}\mathclose{{}\left(ij}\right)}\right), so \mathopen{}\mathclose{{}\left(ji}\right), and it can be shown in the same way as for \mathopen{}\mathclose{{}\left(ij}\right) that \mathopen{}\mathclose{{}\left(ji}\right) is an idempotent. ∎

Proposition 6.5**.**

Let $P$ be a pregroupoid. Then $P$ has a unique preinverses if and only if \mathopen{}\mathclose{{}\left(ij}\right)=\mathopen{}\mathclose{{}\left(ji}\right) for all idempotents $i,j\in P$ such that \mathopen{}\mathclose{{}\left(ij}\right).

Proof.

Let $P$ have unique preinverses. If $i,j\in P$ are idempotents and \mathopen{}\mathclose{{}\left(ij}\right) then \mathopen{}\mathclose{{}\left(ij}\right) and \mathopen{}\mathclose{{}\left(ji}\right) are idempotents by Lemma 6.1. Thus,

[TABLE]

so \mathopen{}\mathclose{{}\left(ij}\right)\,\mathbf{I}\,\mathopen{}\mathclose{{}\left(ji}\right). Hence, \mathopen{}\mathclose{{}\left(ji}\right)\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{\mathopen{}\mathclose{{}\left(ij}\right)}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(ij}\right)^{-1}}\right\}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(ij}\right)}\right\}, using Corollary 5.2.

Conversely, let idempotents in $P$ commute. If $y$ and $y^{\prime}$ are preinverses of $x$, then \mathopen{}\mathclose{{}\left(xyx}\right)=x and \mathopen{}\mathclose{{}\left(xy^{\prime}\!x}\right)=x so that \mathopen{}\mathclose{{}\left(yx}\right) and \mathopen{}\mathclose{{}\left(y^{\prime}\!x}\right), so \mathopen{}\mathclose{{}\left(yx}\right) and \mathopen{}\mathclose{{}\left(y^{\prime}\!x}\right) are idempotents since \mathopen{}\mathclose{{}\left(yxyx}\right)=\mathopen{}\mathclose{{}\left(yx}\right) and \mathopen{}\mathclose{{}\left(y^{\prime}\!xy^{\prime}\!x}\right)=\mathopen{}\mathclose{{}\left(y^{\prime}\!x}\right). Similarly, \mathopen{}\mathclose{{}\left(xy}\right) and \mathopen{}\mathclose{{}\left(xy^{\prime}}\right) are idempotents. We also have \mathopen{}\mathclose{{}\left(yxy}\right)=y and \mathopen{}\mathclose{{}\left(y^{\prime}\!xy^{\prime}}\right)=y^{\prime}. Thus, y\!=\!\mathopen{}\mathclose{{}\left(yxy}\right)=\mathopen{}\mathclose{{}\left(yxy^{\prime}\!xy}\right)=\mathopen{}\mathclose{{}\left(y^{\prime}\!xyxy}\right)=\mathopen{}\mathclose{{}\left(y^{\prime}\!xy}\right)=\mathopen{}\mathclose{{}\left(y^{\prime}\!xy^{\prime}\!xy}\right)=\mathopen{}\mathclose{{}\left(y^{\prime}\!xyxy^{\prime}}\right)=\mathopen{}\mathclose{{}\left(y^{\prime}\!xy^{\prime}}\right)\!=\!y^{\prime}. ∎

Pregroupoids (2)

Proposition 6.6**.**

Let $P$ be a pregroupoid $P$ with unique preinverses, $x\in P$. Then $\lambda_{x}=\lambda_{x}^{-1}$ and $\rho_{x}=\rho_{x}^{-1}$.

Proof.

As $\lambda_{x}$ and $\rho_{x}$ are idempotents by Proposition 6.2, we have $\lambda_{x}=\lambda_{x}^{-1}$ and $\rho_{x}=\rho_{x}^{-1}$ by Corollary 5.2. ∎

Proposition 6.7**.**

Let $P$ be a pregroupoid with unique preinverses, $x\in P$. If $x^{-1}$ is the preinverse of $x$ then \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{\!-1}=x.

Proof.

If $x\,\mathbf{I}\,x^{-1}$ then $x^{-1}\,\mathbf{I}\,x$, so x\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{x^{-1}}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}}\right\}. ∎

Proposition 6.8**.**

Let $P$ be a pregroupoid with unique preinverses, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)^{-1}=\mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right).

Proof.

We have \mathopen{}\mathclose{{}\left(xy}\right), \mathopen{}\mathclose{{}\left(xx^{-1}}\right), \mathopen{}\mathclose{{}\left(x^{-1}x}\right), \mathopen{}\mathclose{{}\left(yy^{-1}}\right) and \mathopen{}\mathclose{{}\left(y^{-1}y}\right). Hence, by Proposition 6.5 and the fact that \mathopen{}\mathclose{{}\left(x^{-1}x}\right) and \mathopen{}\mathclose{{}\left(yy^{-1}}\right) are idempotents,

[TABLE]

so that \mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right), and

[TABLE]

so \mathopen{}\mathclose{{}\left(xy}\right)\,\mathbf{I}\,\mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right), so \mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right)\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(xy}\right)^{-1}}\right\}. ∎

If $P$ is a pregroupoid with unique preinverses then \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}=x, so $P$ can be equipped with a unique bijection

[TABLE]

such that, for all $x\in P$,

[TABLE]

A pregroupoid with unique preinverses can thus be regarded as a prepoloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s},\mathfrak{t}}\right) expanded to a pregroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i,\mathfrak{s},\mathfrak{t}}}\right) characterized by the uniqueness property and the identities (4.1), (4.2), (6.1), and (6.3).

If a pregroupoid with unique preinverses – or alternatively its reduct \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i}}\right) – is a magma, it is thus an inverse semigroup, characterized by the uniqueness of the preinverses and the identities

[TABLE]

Note that a pregroupoid with unique preinverses does not necessarily have unique local left and right units. While \mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \mathopen{}\mathclose{{}\left(x^{-1}x}\right) are uniquely determined by $x$ when its preinverse $x^{-1}$ is unique, and are local left and local right units, respectively, for $x$, \mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \mathopen{}\mathclose{{}\left(x^{-1}x}\right) are not necessarily the only local units for $x$.

Example 6.1**.**

Let $S$ be a set \mathopen{}\mathclose{{}\left\{x,y}\right\} with a binary operation given by the table

[TABLE]

Let \alpha,\beta,\gamma\in\mathopen{}\mathclose{{}\left\{x,y}\right\}. If $\alpha=\beta=\gamma=y$ then \mathopen{}\mathclose{{}\left(\alpha\beta}\right)\gamma=\alpha\mathopen{}\mathclose{{}\left(\beta\gamma}\right)=y; otherwise, \mathopen{}\mathclose{{}\left(\alpha\beta}\right)\gamma=\alpha\mathopen{}\mathclose{{}\left(\beta\gamma}\right)=x. Thus, in all cases \mathopen{}\mathclose{{}\left(\alpha\beta}\right)\gamma=\alpha\mathopen{}\mathclose{{}\left(\beta\gamma}\right), so $S$ is a semigroup. In particular, $xxx=x$ and $yyy=y$, so $x$ is a preinverse of $x$ and $y$ is a preinverse of $y$. Also, $yxy\neq y$, so $x$ is not a preinverse of $y$, and $y$ is not a preinverse of $x$. Hence, $S$ is a pregroupoid where preinverses are unique.

Local units are not unique, however; we have $xx=x$, so $xx^{-1}=x$ is a local left unit for $x$, but $y$ is also a local left unit for $x$ since $yx=x$. As a consequence, if we set \mathfrak{s}_{1}\mathopen{}\mathclose{{}\left(x}\right)=x,\mathfrak{s}_{1}\mathopen{}\mathclose{{}\left(y}\right)=y,\mathfrak{s}_{2}\mathopen{}\mathclose{{}\left(x}\right)=y,\mathfrak{s}_{2}\mathopen{}\mathclose{{}\left(y}\right)=y then \mathopen{}\mathclose{{}\left(\mathfrak{s}_{1}\mathopen{}\mathclose{{}\left(x}\right)x}\right)=\mathopen{}\mathclose{{}\left(\mathfrak{s}_{2}\mathopen{}\mathclose{{}\left(x}\right)x}\right)=x and \mathopen{}\mathclose{{}\left(\mathfrak{s}_{1}\mathopen{}\mathclose{{}\left(y}\right)y}\right)=\mathopen{}\mathclose{{}\left(\mathfrak{s}_{2}\mathopen{}\mathclose{{}\left(y}\right)y}\right)=y but $\mathfrak{s}_{1}\neq\mathfrak{s}_{2}$.

Pregroupoids as prepoloids

Let \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i}}\right) be a reduct of the pregroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i,\mathfrak{s},\mathfrak{t}}}\right). One can expand \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i}}\right) to a pregroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathbf{s},\mathbf{t}}\right) by setting \mathbf{s}\mathopen{}\mathclose{{}\left(x}\right)=\mathopen{}\mathclose{{}\left(x\,\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)}\right) and \mathbf{t}\mathopen{}\mathclose{{}\left(x}\right)=\mathopen{}\mathclose{{}\left(\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)\,x}\right). Finally, we obtain the prepoloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathbf{s},\mathbf{t}}\right) as a reduct of \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i,\mathbf{s},\mathbf{t}}}\right); below we note two useful results about this prepoloid in the case when $\mathbf{s}$ and $\mathbf{t}$ are unique. Recall from Section 4.4 that \boldsymbol{\uplambda}_{x}=\mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \boldsymbol{\uprho}_{x}=\mathopen{}\mathclose{{}\left(x^{-1}x}\right).

Proposition 6.9**.**

Let $P$ be a pregroupoid with unique canonical local units, $x\in P$. Then \mathopen{}\mathclose{{}\left(\boldsymbol{\uplambda}_{x}\boldsymbol{\uplambda}_{x}}\right)=\boldsymbol{\uplambda}_{x}=\boldsymbol{\uplambda}_{\boldsymbol{\uplambda}_{x}}=\boldsymbol{\uprho}{}_{\boldsymbol{\uplambda}_{x}} and \mathopen{}\mathclose{{}\left(\boldsymbol{\uprho}_{x}\boldsymbol{\uprho}_{x}}\right)=\boldsymbol{\uprho}_{x}=\boldsymbol{\uprho}_{\boldsymbol{\uprho}_{x}}=\boldsymbol{\uplambda}{}_{\boldsymbol{\uprho}_{x}}.

Proof.

Analogous to the proof of Proposition 6.2, in addition using the facts that \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=\mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=\mathopen{}\mathclose{{}\left(x^{-1}x}\right). ∎

Proposition 6.10**.**

Let $P$ be a pregroupoid with unique canonical local units, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\boldsymbol{\uplambda}_{x} and \boldsymbol{\uprho}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\boldsymbol{\uprho}_{y}.

Proof.

Analogous to the proof of Proposition 6.3. ∎

6.2. The skew-prepoloid family

Definition 6.2**.**

Let $P$ be a left (resp. right) semigroupoid. Then $P$ is

1. (1)

a left (resp. right) skew-prepoloid when there is a local left (resp. right) unit $\lambda_{x}\in P$ (resp. $\rho_{x}\in P$) for every $x\in P$; 2. (2)

a left (resp. right) skew-pregroupoid when $P$ is a left (resp. right) skew-prepoloid such that for each $x\in P$ there is a right (resp. left) preinverse $x^{-1}$ of $x$ in $P$.

In view of the left-right duality of these notions, it suffices to consider left skew-prepoloids and left skew-pregroupoids here.

By Definition 6.2, a left skew-prepoloid is a left semigroupoid $P$ such that for every $x\in P$ there is some $\lambda{}_{x}\in P$ such that \mathopen{}\mathclose{{}\left(\lambda{}_{x}x}\right)=x. A left skew-groupoid is a left skew-poloid $P$ such that for every $x\in P$ there is some $x^{-1}\in P$ such that \mathopen{}\mathclose{{}\left(xx^{-1}}\right)\!\in\!\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x} and \mathopen{}\mathclose{{}\left(x^{-1}x}\right)\!\in\!\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x^{-1}}, so that \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)\!=\!x and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)x^{-1}}\right)\!=\!x^{-1}.

Left skew-prepoloids

Proposition 6.11**.**

Let $P$ be a left skew-prepoloid, $x,y\in P$. If $\lambda_{y}$ is a local left unit for $y$ and \mathopen{}\mathclose{{}\left(x\lambda_{y}}\right) then \mathopen{}\mathclose{{}\left(xy}\right).

Proof.

If \mathopen{}\mathclose{{}\left(x\lambda_{y}}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\lambda_{y}y}\right)}\right)=\mathopen{}\mathclose{{}\left(xy}\right) since \mathopen{}\mathclose{{}\left(\lambda_{y}y}\right)=y. ∎

Proposition 6.12**.**

Let $P$ be a left skew-prepoloid $P$ with unique local units, $x\in P$. Then \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)=\lambda_{x}=\lambda_{\lambda_{x}}.

Proof.

We have x=\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)=\mathopen{}\mathclose{{}\left(\lambda_{x}\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)x}\right), so \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\lambda_{x}}\right\}, so \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)=\lambda_{x}, so \lambda_{x}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\lambda_{x}}=\mathopen{}\mathclose{{}\left\{\lambda_{\lambda_{x}}}\right\}. ∎

Proposition 6.13**.**

Let $P$ be a left skew-prepoloid $P$ with unique local units, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \lambda_{\mathopen{}\mathclose{{}\left(xy}\right)}=\lambda_{x}.

Proof.

If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)y}\right)=\mathopen{}\mathclose{{}\left(\lambda_{x}\mathopen{}\mathclose{{}\left(xy}\right)}\right), so \lambda_{x}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{\lambda_{\mathopen{}\mathclose{{}\left(xy}\right)}}\right\}. ∎

By Definition 6.2, every left skew-prepoloid $P$ with unique local left units can be equipped with a unique surjective function

[TABLE]

such that, for all $x\in P$,

[TABLE]

By Proposition (6.12), \mathfrak{s}\mathopen{}\mathclose{{}\left(\lambda{}_{x}}\right)=\lambda{}_{x} for all \lambda_{x}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{\!P}.

A left skew-poloid with unique local units can thus be regarded as a left semigroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m}}\right) expanded to a left skew-poloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s}}\right) characterized by the uniqueness property and the identities (4.1) and (6.5).

We call a left semigroupoid $P$ which admits a not necessarily unique function $\mathfrak{s}:P\rightarrow P$ satisfying (6.5) a left unital semigroupoid. If $P$ is a magma then \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s}}\right) is a semigroup such that, for all $x,y,z\in P$,

[TABLE]

Such a semigroup may be called a left unital semigroup. The class of left unital semigroups includes many types of semigroups studied in the literature, for example, $D$-semigroups [16], left abundant semigroups, left adequate semigroups, left Ehresmann semigroups, left ample semigroups and left restriction semigroups.

Left skew-pregroupoids

Proposition 6.14**.**

Let $P$ be a left skew-pregroupoid with unique right preinverses, $x\in P$. Then $\lambda_{x}^{-1}=\lambda_{x}$.

Proof.

As $\lambda_{x}$ is an idempotent by Proposition 6.12, we have \lambda_{x}\in\mathbf{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x}=\mathopen{}\mathclose{{}\left\{\lambda_{x}^{-1}}\right\} as in Corollary 5.2. ∎

Proposition 6.15**.**

Let $P$ be a left skew-pregroupoid with unique right preinverses, $x\in P$. If $x^{-1}$ is the right preinverse of $x$ then x=\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}.

Proof.

If $x\,\mathbf{I}^{+}\,x^{-1}$ then $x^{-1}\,\mathbf{I}^{+}\,x$, so x\in\mathbf{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x^{-1}}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}}\right\}. ∎

If $P$ is a left skew-pregroupoid with unique right preinverses then \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}=x, so $P$ can be equipped with a unique bijection

[TABLE]

such that, for all $x\in P$,

[TABLE]

A left skew-pregroupoid with unique right inverses can thus be regarded as a left skew-poloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s}}\right) expanded to a left skew-groupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathfrak{s}}\right) characterized by the uniqueness property and the identities (4.1), (6.5) and (6.7).

Those left skew-pregroupoids with unique right preinverses which are magmas are again just inverse semigroups, characterized by the uniqueness of preinverses and the identities (6.4).

Skew-pregroupoids as skew-prepoloids

Let \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i}}\right) be a reduct of \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i,\mathfrak{s}}}\right), and expand \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i}}\right) to a left skew-pregroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathbf{s}}\right) by setting \mathbf{s}\mathopen{}\mathclose{{}\left(x}\right)=\mathopen{}\mathclose{{}\left(x\,\mathfrak{i}\mathopen{}\mathclose{{}\left(x}\right)}\right). We note two useful results about the left skew-prepoloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathbf{s}}\right), obtained as a reduct of \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathbf{s}}\right), in the case when $\mathbf{s}$ is unique. Recall that \boldsymbol{\uplambda}_{x}=\mathopen{}\mathclose{{}\left(xx^{-1}}\right).

Proposition 6.16**.**

Let $P$ be a left skew-pregroupoid with unique canonical local left units, $x\in P$. Then \mathopen{}\mathclose{{}\left(\boldsymbol{\uplambda}_{x}\boldsymbol{\uplambda}_{x}}\right)=\boldsymbol{\uplambda}_{x}=\boldsymbol{\uplambda}_{\boldsymbol{\uplambda}{}_{x}}.

Proof.

Analogous to the proof of Proposition 6.12, although note that the present proof requires the fact that \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)x^{-1}}\right)=\mathopen{}\mathclose{{}\left(xx^{-1}}\right) so that \mathopen{}\mathclose{{}\left(\boldsymbol{\uplambda}_{x}\boldsymbol{\uplambda}_{x}}\right)\in\mathopen{}\mathclose{{}\left\{\boldsymbol{\uplambda}}\right\}_{x}. ∎

Proposition 6.17**.**

Let $P$ be a left skew-pregroupoid with unique canonical local left units, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \boldsymbol{\uplambda}_{x}=\boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(xy}\right)}.

Proof.

Analogous to the proof of Proposition 6.13. ∎

7. Poloids and related magmoids

The magmoids considered in this section differ from those in Section 6 in that they are equipped with two-sided or twisted units rather than local units. These magmoids are, roughly speaking, categories and some of their specializations and generalizations, considered as algebraic structures. As noted in the introduction to this article, categories and groupoids are indeed examples of important notions involving partial binary operations.

7.1. The poloid family

Definition 7.1**.**

Let $P$ be a semigroupoid. Then $P$ is

1. (1)

a poloid when there is a left effective unit $\ell_{x}\in P$ and a right effective unit $r_{x}\in P$ for every $x\in P$*; * 2. (2)

a groupoid when $P$ is a poloid such that for every $x\in P$ there is a strong preinverse $x^{-1}\in P$ of $x$.

More explicitly, a poloid is a semigroupoid $P$ such that for every $x\in P$ there are two-sided units $\ell_{x},r_{x}$ such that \mathopen{}\mathclose{{}\left(\ell_{x}x}\right)=\mathopen{}\mathclose{{}\left(xr_{x}}\right)=x, and also \mathopen{}\mathclose{{}\left(y\ell_{x}}\right)=\mathopen{}\mathclose{{}\left(yr_{x}}\right)=\mathopen{}\mathclose{{}\left(\ell_{x}y}\right)=\mathopen{}\mathclose{{}\left(r_{x}y}\right)=y for all $x,y\in P$ such that, respectively, \mathopen{}\mathclose{{}\left(y\ell_{x}}\right), \mathopen{}\mathclose{{}\left(yr_{x}}\right), \mathopen{}\mathclose{{}\left(\ell_{x}y}\right), and \mathopen{}\mathclose{{}\left(r_{x}y}\right). A groupoid is a poloid $P$ such that for every $x\in P$ there is some $x^{-1}\in P$ such that there are two-sided units $\ell_{x},r_{x}$ such that \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=\ell_{x}=r_{x^{-1}} and \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=r_{x}=\ell_{x^{-1}}, so that \mathopen{}\mathclose{{}\left(xx^{-1}x}\right)=x, \mathopen{}\mathclose{{}\left(x^{-1}xx^{-1}}\right)=x^{-1}, and also \mathopen{}\mathclose{{}\left(yxx^{-1}}\right)=\mathopen{}\mathclose{{}\left(yx^{-1}x}\right)=\mathopen{}\mathclose{{}\left(xx^{-1}y}\right)=\mathopen{}\mathclose{{}\left(x^{-1}xy}\right)=y for all $x,y\in P$ such that, respectively, \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right), \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right), \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)y}\right), and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)y}\right).

It is shown below that every element of a poloid has unique left and right effective units and a unique preinverse, and it turns out that poloids have much in common with prepoloids with unique local units and preinverses. A similar remark applies to skew-poloids, defined below, in relation to skew-prepoloids.

A poloid is just a (small) category regarded as an abstract algebraic structure [9], while a groupoid is a (small) category with preinverses, also regarded as an abstract algebraic structure. While categories are usually defined in another way, definitions similar to the definition of poloids given here can also be found in the literature. For example, Ehresmann [2] proposed the following definition:

Eine Kategorie ist eine Klasse $C$ von Elementen, in der eine Multiplikation gegeben ist \mathopen{}\mathclose{{}\left(f,g}\right)\rightarrow fg für gewisse Paare $(f,g)$ von Elementen von $C$, welche folgenden Axiomen genügt:

1. Wenn h\mathopen{}\mathclose{{}\left(fg}\right) oder \mathopen{}\mathclose{{}\left(hf}\right)g definiert ist, dann sind die beide Elemente definiert und h\mathopen{}\mathclose{{}\left(fg}\right)=\mathopen{}\mathclose{{}\left(hf}\right)g.

2. Wenn $hf$ und $fg$ definiert sind, dann ist auch h\mathopen{}\mathclose{{}\left(fg}\right) definiert.

Ein Element $e$ von $C$ wird eine Einheit genannt, falls $fe=f$ und $eg=g$ für alle Elemente $f$ und $g$ von $C$ ist, für welche $fe$ und $eg$ definiert sind.

3. Für jedes $f\in C$ gibt es zwei Einheiten \alpha\mathopen{}\mathclose{{}\left(f}\right) und \beta\mathopen{}\mathclose{{}\left(f}\right), so dass f\alpha\mathopen{}\mathclose{{}\left(f}\right) und \beta\mathopen{}\mathclose{{}\left(f}\right)f definiert sind. (p. 50).

Proposition 7.1 below implies that if a poloid is a semigroup then it is a monoid, since it has only one two-sided unit, denoted $1$, and if a groupoid is a semigroup then it is a group. Conversely, a poloid with just one two-sided unit is a monoid, and a groupoid with just one two-sided unit is a group [9]. A groupoid is thus a generalized group, as expected, while poloids generalize groups indirectly and in two ways, via monoids and via groupoids.

Poloids

Proposition 7.1**.**

Let $P$ be a poloid. If e,e^{\prime}\in\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P} and \mathopen{}\mathclose{{}\left(ee^{\prime}}\right) then $e=e^{\prime}$.

Proof.

We have e=\mathopen{}\mathclose{{}\left(ee^{\prime}}\right)=e^{\prime}. ∎

Proposition 7.2**.**

Let $P$ be a poloid, $x\in P$. Then there is a unique left effective unit $\ell_{x}\in P$ and a unique right effective unit $r_{x}\in P$ for every $x\in P$.

Proof.

If \ell_{x},\ell_{x}^{\prime}\in\mathopen{}\mathclose{{}\left\{\ell}\right\}_{x} then x=\mathopen{}\mathclose{{}\left(\ell_{x}x}\right)=\mathopen{}\mathclose{{}\left(\ell_{x}\mathopen{}\mathclose{{}\left(\ell_{x}^{\prime}x}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\ell_{x}\ell_{x}^{\prime}}\right)x}\right), so \mathopen{}\mathclose{{}\left(\ell_{x}\ell_{x}^{\prime}}\right) and thus $\ell_{x}=\ell_{x}^{\prime}$. Dually, if r_{x},r_{x}^{\prime}\in\mathopen{}\mathclose{{}\left\{r}\right\}_{x} then x=\mathopen{}\mathclose{{}\left(xr_{x}}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xr_{x}^{\prime}}\right)r_{x}}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(r_{x}^{\prime}r_{x}}\right)}\right), so \mathopen{}\mathclose{{}\left(r_{x}^{\prime}r_{x}}\right) and thus $r_{x}=r_{x}^{\prime}$. ∎

Proposition 7.3**.**

Let $P$ be a poloid, $x,y\in P$. Then \mathopen{}\mathclose{{}\left(xy}\right) if and only if $r_{x}=\ell_{y}$.

Proof.

If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xr_{x}}\right)\mathopen{}\mathclose{{}\left(\ell_{y}y}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xr_{x}}\right)\ell_{y}}\right)y}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(r_{x}\ell_{y}}\right)}\right)y}\right), so \mathopen{}\mathclose{{}\left(r_{x}\ell_{y}}\right), so $r_{x}=\ell{}_{y}$. Conversely, if $r_{x}=\ell{}_{y}$ then \mathopen{}\mathclose{{}\left(x\ell_{y}}\right), so \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\ell_{y}y}\right)}\right) and thus \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\ell_{y}y}\right)}\right)=\mathopen{}\mathclose{{}\left(xy}\right) since \mathopen{}\mathclose{{}\left(\ell_{y}y}\right)=y. ∎

Proposition 7.4**.**

Let $P$ be a poloid. If e\in\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P} then \mathopen{}\mathclose{{}\left(ee}\right)=e=\ell_{e}=r_{e}.

Proof.

We have e=\mathopen{}\mathclose{{}\left(\ell{}_{e}e}\right)=\ell{}_{e} and e=\mathopen{}\mathclose{{}\left(er_{e}}\right)=r_{e}. ∎

Corollary 7.1**.**

Let $P$ be a poloid, $x\in P$. Then \mathopen{}\mathclose{{}\left(\ell{}_{x}\ell{}_{x}}\right)=\ell_{x}=\ell_{\ell_{x}}=r{}_{\ell_{x}} and \mathopen{}\mathclose{{}\left(r_{x}r_{x}}\right)=r_{x}=\ell{}_{r_{x}}=r_{r_{x}}.

Proposition 7.5**.**

Let $P$ be a poloid, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \ell_{\mathopen{}\mathclose{{}\left(xy}\right)}\!=\!\ell_{x} and r_{\mathopen{}\mathclose{{}\left(xy}\right)}\!=\!r_{y}.

Proof.

We use the fact that effective units are unique by Proposition 7.2. If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\ell_{x}x}\right)y}\right)=\mathopen{}\mathclose{{}\left(\ell_{x}\mathopen{}\mathclose{{}\left(xy}\right)}\right), so \ell_{x}\in\mathopen{}\mathclose{{}\left\{\ell}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{\ell_{\mathopen{}\mathclose{{}\left(xy}\right)}}\right\}. Dually, \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yr_{y}}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)r_{y}}\right), so r_{y}\in\mathopen{}\mathclose{{}\left\{r}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{r_{\mathopen{}\mathclose{{}\left(xy}\right)}}\right\}. ∎

In view of Definition (7.1) and Proposition 7.2, every poloid $P$ can be equipped with unique surjective functions

[TABLE]

such that, for any $x,y\in P$,

[TABLE]

By Corollary 7.1, \mathfrak{s}\mathopen{}\mathclose{{}\left(\ell_{x}}\right)=\ell_{x} for all \ell_{x}\in\mathopen{}\mathclose{{}\left\{\ell}\right\}_{\!P} and \mathfrak{t}\mathopen{}\mathclose{{}\left(r_{x}}\right)=r_{x} for all r_{x}\in\mathopen{}\mathclose{{}\left\{r}\right\}_{\!P}.

A poloid $P$ can thus be regarded as an expansion \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s},\mathfrak{t}}\right), characterized by the identities (4.1), (4.2), and (7.1), of a semigroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m}}\right).

If \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s},\mathfrak{t}}\right) is a magma then it degenerates to a monoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},1}\right) where \mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)=\mathfrak{t}\mathopen{}\mathclose{{}\left(x}\right)=1 for all $x\in P$.

Groupoids

Proposition 7.6**.**

Let $P$ be a poloid, $x\in P$. If x^{\mathopen{}\mathclose{{}\left(-1}\right)} is a strong pseudoinverse of $x$ then x^{\mathopen{}\mathclose{{}\left(-1}\right)} is a strong preinverse of $x$.

Proof.

For any strong pseudoinverse x^{\mathopen{}\mathclose{{}\left(-1}\right)} of $x$, \mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(-1}\right)}}\right)\in\mathopen{}\mathclose{{}\left\{\ell}\right\}_{x}\subseteq\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P} and \mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(-1}\right)}x}\right)\in\mathopen{}\mathclose{{}\left\{r}\right\}_{x}\subseteq\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P}, so \mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(-1}\right)}\mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(-1}\right)}}\right)}\right)=x^{\mathopen{}\mathclose{{}\left(-1}\right)} and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(-1}\right)}x}\right)x^{\mathopen{}\mathclose{{}\left(-1}\right)}}\right)=x^{\mathopen{}\mathclose{{}\left(-1}\right)}. Thus, \mathopen{}\mathclose{{}\left(xx^{\mathopen{}\mathclose{{}\left(-1}\right)}}\right)\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x^{-1}} and \mathopen{}\mathclose{{}\left(x^{\mathopen{}\mathclose{{}\left(-1}\right)}x}\right)\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{x^{-1}}, so x^{\mathopen{}\mathclose{{}\left(-1}\right)} is a preinverse of $x$, and hence a strong preinverse of $x$. ∎

Hence, one may alternatively define a groupoid as a poloid $P$ such that for every $x\in P$ there is a strong pseudoinverse x^{\mathopen{}\mathclose{{}\left(-1}\right)}\in P of $x$, and definitions of this form are common in the literature.

Proposition 7.7**.**

Let $P$ be a poloid, $x\in P$. Then there is at most one strong pseudoinverse x^{\mathopen{}\mathclose{{}\left(-1}\right)}\in P of $x$.

Proof.

If $x^{\prime}$ and $x^{\prime\prime}$ are strong pseudoinverses of $x$ then \mathopen{}\mathclose{{}\left(xx^{\prime}}\right)\in\mathopen{}\mathclose{{}\left\{\ell}\right\}_{x}\subseteq\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P} and \mathopen{}\mathclose{{}\left(x^{\prime\prime}\!x}\right)\in\mathopen{}\mathclose{{}\left\{r}\right\}_{x}\subseteq\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P}, so x^{\prime}=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{\prime\prime}x}\right)x^{\prime}}\right)=\mathopen{}\mathclose{{}\left(x^{\prime\prime}\mathopen{}\mathclose{{}\left(xx^{\prime}}\right)}\right)=x^{\prime\prime}. ∎

Corollary 7.2**.**

Let $P$ be a groupoid. Then every $x\in P$ has a unique strong preinverse $x^{-1}\in P$.

Proposition 7.8**.**

Let $P$ be a groupoid. If $e\in P$ is a two-sided unit then $e^{-1}=e$.

Proof.

The assertion is that \mathopen{}\mathclose{{}\left\{e}\right\}=\boldsymbol{I}\mathopen{}\mathclose{{}\left\{}\right\}_{e}. As \mathopen{}\mathclose{{}\left(ee}\right)=e we have \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(ee}\right)e}\right)=\mathopen{}\mathclose{{}\left(e\mathopen{}\mathclose{{}\left(ee}\right)}\right)=e, so $e\,\mathbf{I}\,e$, so e\in\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{e}=\mathopen{}\mathclose{{}\left\{e^{-1}}\right\} since preinverses are unique. Thus, \mathopen{}\mathclose{{}\left\{e}\right\}=\mathbf{I}\mathopen{}\mathclose{{}\left\{}\right\}_{e}, so to prove the assertion it suffices to note that \mathopen{}\mathclose{{}\left(ee^{-1}}\right)=\mathopen{}\mathclose{{}\left(e^{-1}e}\right)=\mathopen{}\mathclose{{}\left(ee}\right)=e, so that \mathopen{}\mathclose{{}\left(ee^{-1}}\right) and \mathopen{}\mathclose{{}\left(e^{-1}e}\right) are two-sided units. ∎

Proposition 7.9**.**

Let $P$ be a groupoid, $x\in P$. Then \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}=x.

Proof.

If $x\,\mathit{\boldsymbol{I}}\,x^{-1}$ then $x^{-1}\,\boldsymbol{I}\,x$, meaning that $x$ is a strong preinverse of $x^{-1}$. Thus, x\in\boldsymbol{I}\mathopen{}\mathclose{{}\left\{}\right\}_{x^{-1}}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}}\right\} since strong preinverses are unique. ∎

Proposition 7.10**.**

Let $P$ be a groupoid, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)^{-1}=\mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right).

Proof.

If \mathopen{}\mathclose{{}\left(xy}\right) then $r_{x}=\ell_{x^{-1}}=\ell_{y}=r_{y^{-1}}$, so \mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right). Furthermore, we have \mathopen{}\mathclose{{}\left(x^{-1}x}\right)\in\mathopen{}\mathclose{{}\left\{r_{x}}\right\}\subseteq\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P} and \mathopen{}\mathclose{{}\left(yy^{-1}}\right)\in\mathopen{}\mathclose{{}\left\{\ell_{y}}\right\}\subseteq\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P}, so \mathopen{}\mathclose{{}\left(xyy^{-1}x^{-1}xy}\right), and we obtain

[TABLE]

Also, \mathopen{}\mathclose{{}\left(y^{-1}x^{-1}xyy^{-1}x^{-1}}\right), and we have

[TABLE]

Thus, \mathopen{}\mathclose{{}\left(xy}\right)\,\mathbf{I}\,\mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right). Also,

[TABLE]

so \mathopen{}\mathclose{{}\left(xy}\right)\,\boldsymbol{I}\,\mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right). Hence, \mathopen{}\mathclose{{}\left(y^{-1}x^{-1}}\right)\in\boldsymbol{I}\mathopen{}\mathclose{{}\left\{}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(xy}\right)^{-1}}\right\} since strong preinverses are unique. ∎

Since every $x\in P$ has a unique preinverse $x^{-1}$ such that \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}=x, there is a unique bijection

[TABLE]

such that, for any $x,y\in P$,

[TABLE]

A groupoid $P$ can thus be regarded as an expansion \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathfrak{s},\mathfrak{t}}\right), characterized by the identities (4.1), (4.2), (7.1), and (7.2), of a poloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s},\mathfrak{t}}\right),

If \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathfrak{s},\mathfrak{t}}\right) is a magma then it degenerates to a group \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},1}\right) where \mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)=\mathfrak{t}\mathopen{}\mathclose{{}\left(x}\right)=1 for all $x\in P$.

7.2. The skew-poloid family

Definition 7.2**.**

Let $P$ be a left (resp. right) semigroupoid. Then $P$ is

1. (1)

a left (resp. right) skew-poloid when there is a unique twisted left (resp. right) unit $\varphi_{x}\in P$ (resp. $\psi_{x}\in P$) for every $x\in P$; 2. (2)

a left (resp. right) skew-groupoid when $P$ is a left (resp. right) skew-poloid such that for every $x\in P$ there is a strong left (resp. right) preinverse $x^{-1}\in P$ of $x$ and a unique $\varphi_{x^{-1}}\in P$ (resp. $\psi_{x^{-1}}\in P$) such that if $x^{-1}$ is a strong left (resp. right) preinverse of $x$ then $\varphi_{x^{-1}}$ (resp. $\psi_{x^{-1}}$) is the twisted left (resp. right) unit for $x^{-1}$.

In view of the left-right duality in the skew-poloid family, it suffices to consider only left skew-poloids and left skew-groupoids below. Note that we cannot in general regard \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) and \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) as equivalent expressions, written \mathopen{}\mathclose{{}\left(xyz}\right), in this case; \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) implies \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right), but not conversely.

By Definition 7.2, a left skew-poloid is a left semigroupoid $P$ such that for every $x\in P$ there is a unique $\varphi_{x}\in P$ such that \mathopen{}\mathclose{{}\left(\varphi_{x}x}\right)=x and such that \mathopen{}\mathclose{{}\left(y\varphi_{x}}\right)=y for every $y\in P$ such that \mathopen{}\mathclose{{}\left(y\varphi_{x}}\right). A left skew-groupoid is a left skew-poloid $P$ such that for every $x\in P$ there is some $x^{-1}\in P$ such that \varphi_{x}=\mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \varphi{}_{x^{-1}}=\mathopen{}\mathclose{{}\left(x^{-1}x}\right), so that \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)=x, \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}x}\right)x^{-1}}\right)=x^{-1}, \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right)=y for all $x,y\in P$ such that \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(xx^{-1}}\right)}\right), and \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=y for all $x,y\in P$ such that \mathopen{}\mathclose{{}\left(y\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right). In addition, because of the two uniqueness assumptions in Definition 7.2, we have that if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{\prime}}\right)x}\right)\!=\!x, \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{\prime}x}\right)x^{\prime}}\right)\!=\!x^{\prime} and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{\prime\prime}}\right)x}\right)=x,\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{\prime\prime}x}\right)x^{\prime\prime}}\right)=x^{\prime\prime} then not only \mathopen{}\mathclose{{}\left(xx^{\prime}}\right)=\mathopen{}\mathclose{{}\left(xx^{\prime\prime}}\right) but also \mathopen{}\mathclose{{}\left(x^{\prime}x}\right)=\mathopen{}\mathclose{{}\left(x^{\prime\prime}x}\right).

A left (or one-sided) skew-poloid is what has been called a constellation [6, 4, 5]. There is a close relationship between poloids and (left) skew-poloids, or between categories and constellations, because both notions formalize the idea of a system of (structured) sets and many-to-one correspondences between these sets. Without going into details, the difference between the two notions is that in the first case many-to-one correspondences are formalized as functions, with domains and codomains, whereas in the second case, many-to-one correspondences are formalized as prefunctions, with domains but without codomains.888For details, see [9, 5]. In [5], prefunctions are interpreted as surjective functions; the prefunction $\mathsf{f:}X\rightarrow Y$ is rendered as the function $f:X\rightarrow\mathrm{im}\,f\subseteq Y$.

Left skew-poloids

Proposition 7.11**.**

Let $P$ be a left skew-poloid. If \varphi,\varphi^{\prime}\in\mathopen{}\mathclose{{}\left\{\varphi}\right\}_{\!P} and \mathopen{}\mathclose{{}\left(\varphi\varphi^{\prime}}\right)=\mathopen{}\mathclose{{}\left(\varphi^{\prime}\varphi}\right) then $\varphi=\varphi^{\prime}$.

Proof.

We have \varphi=\mathopen{}\mathclose{{}\left(\varphi\varphi^{\prime}}\right)=\mathopen{}\mathclose{{}\left(\varphi^{\prime}\varphi}\right)=\varphi^{\prime}. ∎

Proposition 7.12**.**

Let $P$ be a left skew-poloid, $x,y\in P$. Then \mathopen{}\mathclose{{}\left(xy}\right) if and only if \mathopen{}\mathclose{{}\left(x\varphi_{y}}\right).

Proof.

If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\varphi_{y}y}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\varphi_{y}}\right)y}\right), and if \mathopen{}\mathclose{{}\left(x\varphi_{y}}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(\varphi_{y}y}\right)}\right)=\mathopen{}\mathclose{{}\left(xy}\right) since \mathopen{}\mathclose{{}\left(\varphi_{y}y}\right)=y. ∎

Corollary 7.3**.**

Let $P$ be a left skew-poloid, $x,y\in P$. Then \mathopen{}\mathclose{{}\left(xy}\right) if and only if \mathopen{}\mathclose{{}\left(x\varphi_{y}}\right)=x.

Proposition 7.13**.**

Let $P$ be a left skew-poloid, $x\in P$. Then \mathopen{}\mathclose{{}\left(\varphi_{x}\varphi_{x}}\right)=\varphi_{x}=\varphi_{\varphi_{x}}.

Proof.

We have \varphi_{\varphi_{x}}=\mathopen{}\mathclose{{}\left(\varphi_{\varphi_{x}}\varphi_{x}}\right)=\varphi_{x}. ∎

Proposition 7.14**.**

Let $P$ be a left skew-poloid, $x,y\in P$. If \mathopen{}\mathclose{{}\left(xy}\right) then \varphi{}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\varphi_{x}.

Proof.

If \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\varphi_{x}x}\right)y}\right)=\mathopen{}\mathclose{{}\left(\varphi_{x}\mathopen{}\mathclose{{}\left(xy}\right)}\right), so \varphi_{x}\in\mathopen{}\mathclose{{}\left\{\varphi}\right\}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\mathopen{}\mathclose{{}\left\{\varphi_{\mathopen{}\mathclose{{}\left(xy}\right)}}\right\}. ∎

In view of Definition 7.2, every left skew-poloid $P$ can be equipped with a unique surjective function

[TABLE]

such that, for any $x,y\in P$,

[TABLE]

By Proposition 7.13, \mathfrak{s}\mathopen{}\mathclose{{}\left(\varphi{}_{x}}\right)=\varphi{}_{x} for all \varphi{}_{x}\in\mathopen{}\mathclose{{}\left\{\varphi}\right\}_{\!P}.

A left skew-poloid can thus be regarded as an expansion \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s}}\right), characterized by the identities (4.1) and (7.3), of a left semigroupoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m}}\right).

If \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s}}\right) is a magma then it degenerates to a monoid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},1}\right) where \mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)=\mathfrak{t}\mathopen{}\mathclose{{}\left(x}\right)=1 for all $x\in P$.999If \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) then \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) since \mathopen{}\mathclose{{}\left(yz}\right), so if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right) or \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right) then \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right).

Left skew-groupoids

Proposition 7.15**.**

Let $P$ be a left skew-groupoid, $x\in P$. Then there is at most one strong right preinverse $x^{-1}\in P$ of $x$.

Proof.

Let $x^{\prime}$ and $x^{\prime\prime}$ be strong right preinverses of $x$. By the uniqueness of $\varphi_{x}$, $\varphi_{x^{\prime}}$, and $\varphi_{x^{\prime\prime}}$ for all $x\in P$, we have \varphi_{x}=\mathopen{}\mathclose{{}\left(xx^{\prime}}\right)=\mathopen{}\mathclose{{}\left(xx^{\prime\prime}}\right) as well as \varphi_{x^{\prime}}=\mathopen{}\mathclose{{}\left(x^{\prime}x}\right)=\mathopen{}\mathclose{{}\left(x^{\prime\prime}x}\right)=\varphi_{x^{\prime\prime}}. Thus, x^{\prime}\!=\!\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{\prime}x}\right)x^{\prime}}\right)\!=\!\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{\prime\prime}x}\right)x^{\prime}}\right)=\!\mathopen{}\mathclose{{}\left(x^{\prime\prime}\mathopen{}\mathclose{{}\left(xx^{\prime}}\right)}\right)\!=\!\mathopen{}\mathclose{{}\left(x^{\prime\prime}\mathopen{}\mathclose{{}\left(xx^{\prime\prime}}\right)}\right)\!=\!\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{\prime\prime}x}\right)x^{\prime\prime}}\right)\!=\!x^{\prime\prime}. ∎

Corollary 7.4**.**

Let $P$ be a left skew-groupoid, $x\in P$. Then $x$ has a unique strong right preinverse $x^{-1}\in P$.

Proposition 7.16**.**

Let $P$ be a left skew-groupoid. If \varphi\in\mathopen{}\mathclose{{}\left\{\varphi}\right\}_{\!P} then $\varphi^{-1}=\varphi$.

Proof.

The assertion is that \mathopen{}\mathclose{{}\left\{\varphi}\right\}=\boldsymbol{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{\varphi}. By Proposition 7.13, \mathopen{}\mathclose{{}\left(\varphi\varphi}\right)=\varphi, so \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\varphi\varphi}\right)\varphi}\right)=\varphi, so $\varphi\,\mathbf{I}^{+}\varphi$, so \varphi\in\mathbf{I}^{+}\mathopen{}\mathclose{{}\left\{}\right\}_{\varphi}=\mathopen{}\mathclose{{}\left\{\varphi^{-1}}\right\} since strong right preinverses are unique. Thus, \mathopen{}\mathclose{{}\left\{\varphi}\right\}=\mathbf{I}^{+}\mathopen{}\mathclose{{}\left\{}\right\}_{\varphi}, so to prove the assertion it suffices to note that \mathopen{}\mathclose{{}\left(\varphi\varphi^{-1}}\right)=\mathopen{}\mathclose{{}\left(\varphi^{-1}\varphi}\right)=\mathopen{}\mathclose{{}\left(\varphi\varphi}\right)=\varphi, so that \mathopen{}\mathclose{{}\left(\varphi\varphi^{-1}}\right),\mathopen{}\mathclose{{}\left(\varphi^{-1}\varphi}\right)\in\mathopen{}\mathclose{{}\left\{\varepsilon}\right\}_{\!P}. ∎

Proposition 7.17**.**

Let $P$ be a left skew-groupoid, $x\in P$. Then \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}=x.

Proof.

If $x\,\boldsymbol{I}{}^{+}x^{-1}$ then $x^{-1}\,\boldsymbol{I}{}^{+}x$, meaning that $x$ is a strong right preinverse of $x^{-1}$. Thus, x\in\boldsymbol{I}^{+}\!\mathopen{}\mathclose{{}\left\{}\right\}_{x^{-1}}=\mathopen{}\mathclose{{}\left\{\mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}}\right\} since strong right preinverses are unique. ∎

Since every $x\in P$ has a unique preinverse $x^{-1}$ such that \mathopen{}\mathclose{{}\left(x^{-1}}\right)^{-1}=x, there is a unique bijection

[TABLE]

such that, for all $x,y\in P$,

[TABLE]

A left skew-groupoid can be thus be regarded as an expansion \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathfrak{s}}\right), characterized by the identities (4.1), (7.3), and (7.4), of a left skew-poloid \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{s}}\right).

If \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},\mathfrak{s}}\right) is a magma then it degenerates to a group \mathopen{}\mathclose{{}\left(P,\mathfrak{m},\mathfrak{i},1}\right) where \mathfrak{s}\mathopen{}\mathclose{{}\left(x}\right)=1 for all $x\in P$.

8. Prepoloids and pregroupoids

with restricted binary operations

In a poloid, \mathopen{}\mathclose{{}\left(xy}\right) if and only if $r_{x}=\ell_{y}$. In a prepoloid, $\rho_{x}=\lambda_{y}$ implies \mathopen{}\mathclose{{}\left(xy}\right) by Proposition 6.1, but $\rho_{x}=\lambda_{y}$ is not a necessary condition for \mathopen{}\mathclose{{}\left(xy}\right). If we retain only those products \mathopen{}\mathclose{{}\left(xy}\right) for which $\rho_{x}=\lambda_{y}$, we obtain a magmoid P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] with the same elements as $P$ but restricted multiplication \boldsymbol{\mathsf{m}}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(x\cdot y}\right). By definition, we then have \mathopen{}\mathclose{{}\left(x\cdot y}\right) if and only if $\rho_{x}=\lambda_{y}$, as in a poloid, and under suitable conditions the restricted magmoid does indeed become a poloid.

By similarly restricting the binary operation, we can derive a groupoid from a pregroupoid, a skew-poloid from a skew-prepoloid and a skew-groupoid from a skew-pregroupoid.

8.1. From prepoloids to poloids

Definition 8.1**.**

Let $P$ be* a prepoloid with binary operation \mathfrak{m}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(xy}\right), $x,y\in P$. The restricted binary operation* on the carrier set of $P$ is a binary operation \boldsymbol{\mathsf{m}}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(x\cdot y}\right) such that if \mathopen{}\mathclose{{}\left(x\cdot y}\right) and \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right), and \mathopen{}\mathclose{{}\left(x\cdot y}\right) if and only if relative to $\mathfrak{m}$ there is some \rho_{x}\in\mathopen{}\mathclose{{}\left\{\rho}\right\}_{x} and some \lambda_{y}\in\mathopen{}\mathclose{{}\left\{\lambda}\right\}_{y} such that $\rho_{x}=\lambda_{y}$.

In particular, Definition 8.1 applies to magmoids where $\lambda_{x}$ and $\rho_{x}$ are unique local units for $x\in P$.

Note that if \mathopen{}\mathclose{{}\left(x\cdot y}\right) then $\rho_{x}=\lambda_{y}$, so \mathopen{}\mathclose{{}\left(xy}\right), so if \mathopen{}\mathclose{{}\left(x\cdot y}\right) then \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right) by Definition 8.1.

Lemma 8.1**.**

Let $P$ be a prepoloid with unique local units. If $\lambda{}_{x}=\lambda{}_{\lambda_{x}}=\rho_{\lambda_{x}}$ and $\rho_{x}=\lambda_{\rho_{x}}=\rho_{\rho_{x}}$ for all $x\in P$ and also \lambda_{\mathopen{}\mathclose{{}\left(xy}\right)}=\lambda_{x} and \rho_{\mathopen{}\mathclose{{}\left(xy}\right)}=\rho_{y} for all $x,y\in P$ such that \mathopen{}\mathclose{{}\left(xy}\right) then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a poloid where $\ell_{x}=\lambda_{x}$ and $r_{x}=\rho_{x}$ for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right].

Proof.

We first prove that P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a semigroupoid. If \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right) then \rho_{x}=\lambda_{\mathopen{}\mathclose{{}\left(y\cdot z}\right)} and $\rho_{y}=\lambda_{z}$, so \rho_{x}=\lambda_{\mathopen{}\mathclose{{}\left(y\cdot z}\right)}=\lambda_{\mathopen{}\mathclose{{}\left(yz}\right)}=\lambda_{y}. Thus, \rho_{\mathopen{}\mathclose{{}\left(x\cdot y}\right)}=\rho_{\mathopen{}\mathclose{{}\left(xy}\right)}=\rho_{y}=\lambda_{z}, so \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right), so \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right).

Dually, if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right) then \rho_{\mathopen{}\mathclose{{}\left(x\cdot y}\right)}=\lambda_{z} and $\rho_{x}=\lambda_{y}$, so \rho_{y}=\rho_{\mathopen{}\mathclose{{}\left(xy}\right)}=\rho_{\mathopen{}\mathclose{{}\left(x\cdot y}\right)}=\lambda_{z}. Thus, \rho_{x}=\lambda_{y}=\lambda_{\mathopen{}\mathclose{{}\left(yz}\right)}=\lambda_{\mathopen{}\mathclose{{}\left(y\cdot z}\right)}, so \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right), so \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right).

Also, if \mathopen{}\mathclose{{}\left(x\cdot y}\right) and \mathopen{}\mathclose{{}\left(y\cdot z}\right) then $\rho_{x}=\lambda_{y}$ and $\rho_{y}=\lambda_{z}$, so \rho_{x}=\lambda_{y}=\lambda_{\mathopen{}\mathclose{{}\left(yz}\right)}=\lambda_{\mathopen{}\mathclose{{}\left(y\cdot z}\right)} and \rho_{\mathopen{}\mathclose{{}\left(x\cdot y}\right)}=\rho_{\mathopen{}\mathclose{{}\left(xy}\right)}=\rho_{y}=\lambda_{z}. Hence, \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right) and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right), so \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right).

It remains to show that $\lambda_{x}$ and $\rho_{x}$ are two-sided units such that \mathopen{}\mathclose{{}\left(\lambda{}_{x}\cdot x}\right)=x and \mathopen{}\mathclose{{}\left(x\cdot\rho{}_{x}}\right)=x for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right]. If \mathopen{}\mathclose{{}\left(\lambda_{x}\cdot y}\right) then $\lambda{}_{x}=\rho_{\lambda_{x}}=\lambda{}_{y}$, so \mathopen{}\mathclose{{}\left(\lambda_{x}\cdot y}\right)=\mathopen{}\mathclose{{}\left(\lambda_{x}y}\right)=\mathopen{}\mathclose{{}\left(\lambda_{y}y}\right)=y, and if \mathopen{}\mathclose{{}\left(y\cdot\lambda{}_{x}}\right) then $\rho_{y}=\lambda{}_{\lambda_{x}}=\lambda{}_{x}$, so \mathopen{}\mathclose{{}\left(y\cdot\lambda{}_{x}}\right)=\mathopen{}\mathclose{{}\left(y\lambda{}_{x}}\right)=\mathopen{}\mathclose{{}\left(y\rho_{y}}\right)=y.

Similarly, if \mathopen{}\mathclose{{}\left(\rho_{x}\cdot y}\right) then $\rho_{x}=\rho_{\rho_{x}}=\lambda_{y}$, so \mathopen{}\mathclose{{}\left(\rho_{x}\cdot y}\right)=\mathopen{}\mathclose{{}\left(\rho_{x}y}\right)=\mathopen{}\mathclose{{}\left(\lambda_{y}y}\right)=y, and if \mathopen{}\mathclose{{}\left(y\cdot\rho_{x}}\right) then $\rho_{y}=\lambda_{\rho_{x}}=\rho_{x}$ so \mathopen{}\mathclose{{}\left(y\cdot\rho_{x}}\right)=\mathopen{}\mathclose{{}\left(y\rho_{x}}\right)=\mathopen{}\mathclose{{}\left(y\rho_{y}}\right)=y.

Thus, $\lambda_{x}$ and $\rho_{x}$ are two-sided units in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right], and we have \mathopen{}\mathclose{{}\left(\lambda_{x}\cdot x}\right)=\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)=x and \mathopen{}\mathclose{{}\left(x\cdot\rho{}_{x}}\right)=\mathopen{}\mathclose{{}\left(x\rho{}_{x}}\right)=x for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right]. ∎

Combining Lemma 8.1 with Propositions 6.2 and 6.3 we obtain the following results:

Theorem 8.1**.**

If $P$ is a prepoloid with unique local units then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a poloid.

Corollary 8.1**.**

If $P$ is a bi-unital semigroup with unique local units then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a poloid.

8.2. From pregroupoids to groupoids

Definition 8.2**.**

Let $P$ be* a pregroupoid with binary operation \mathfrak{m}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(xy}\right), $x,y\in P$. The restricted binary operation *on the carrier set of $P$ is a binary operation \boldsymbol{\boldsymbol{\mathsf{m}}}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(x\cdot y}\right) such that if \mathopen{}\mathclose{{}\left(x\cdot y}\right) and \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right), and \mathopen{}\mathclose{{}\left(x\cdot y}\right) if and only if there is a canonical local right unit \boldsymbol{\uprho}_{x}=\mathopen{}\mathclose{{}\left(x^{-1}x}\right) for $x$ and a canonical local left unit \boldsymbol{\uplambda}_{y}=\mathopen{}\mathclose{{}\left(yy^{-1}}\right) for $y$ such that $\boldsymbol{\uprho}_{x}=\boldsymbol{\uplambda}_{y}$.

If \mathopen{}\mathclose{{}\left(x\cdot y}\right) then there are $\boldsymbol{\uprho}_{x},\boldsymbol{\uplambda}_{y}$ such that $\boldsymbol{\uprho}_{x}=\boldsymbol{\uplambda}_{y}$, so \mathopen{}\mathclose{{}\left(xy}\right) by Proposition 6.1, so \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right).

Lemma 8.2**.**

Let $P$ be a pregroupoid with unique canonical local units. If $\boldsymbol{\uplambda}{}_{x}=\boldsymbol{\uplambda}{}_{\boldsymbol{\uplambda}_{x}}=\boldsymbol{\uprho}_{\lambda_{x}}$ and $\boldsymbol{\uprho}_{x}=\boldsymbol{\uplambda}_{\boldsymbol{\uprho}_{x}}=\boldsymbol{\uprho}_{\boldsymbol{\uprho}_{x}}$ for all $x\in P$ and also \boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\boldsymbol{\uplambda}_{x} and \boldsymbol{\uprho}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\boldsymbol{\uprho}_{y} for all $x,y\in P$ such that \mathopen{}\mathclose{{}\left(xy}\right) then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a groupoid where $\ell_{x}=\boldsymbol{\uplambda}{}_{x}$ and $r_{x}=\boldsymbol{\uprho}_{x}$ for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right].

Proof.

We can prove that P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a poloid where $\ell_{x}=\boldsymbol{\uplambda}{}_{x}$ and $r_{x}=\boldsymbol{\uprho}_{x}$ for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] by using the argument in the proof of Lemma 8.1 again. It remains to show that every x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] has a strong preinverse in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right]. Let $x^{-1}$ be a preinverse of $x$ in $P$. As \boldsymbol{\uprho}_{x}=\mathopen{}\mathclose{{}\left(x^{-1}x}\right)=\boldsymbol{\lambda}_{x^{-1}} we have \mathopen{}\mathclose{{}\left(x\cdot x^{-1}}\right), and as \boldsymbol{\uprho}_{x^{-1}}=\mathopen{}\mathclose{{}\left(xx^{-1}}\right)=\boldsymbol{\lambda}_{x} we have \mathopen{}\mathclose{{}\left(x^{-1}\cdot x}\right), so \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot x^{-1}}\right)\cdot x}\right) and \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(x^{-1}\cdot x}\right)}\right). Thus, \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot x^{-1}}\right)\cdot x}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xx^{-1}}\right)x}\right)=x and \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(x^{-1}\cdot x}\right)}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(x^{-1}x}\right)}\right)=x, so \mathopen{}\mathclose{{}\left(x\cdot x^{-1}}\right)=\ell_{x}\in\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P} and \mathopen{}\mathclose{{}\left(x^{-1}\cdot x}\right)=r_{x}\in\mathopen{}\mathclose{{}\left\{e}\right\}_{\!P} by the uniqueness of effective units in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right]. Thus $x^{-1}$ is a strong pseudoinverse of $x$ in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] and hence a strong preinverse of $x$ in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] by Proposition 7.6. ∎

Combining Lemma 8.2 with Propositions 6.9 and 6.10, we obtain the following results:

Theorem 8.2**.**

If $P$ is a pregroupoid with unique canonical local units then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathfrak{m}}}\right] is a groupoid.

Corollary 8.2**.**

If $S$ is a regular semigroup with unique canonical local units then S\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a groupoid.

Theorem 8.3**.**

If $P$ is a pregroupoid with unique preinverses then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a groupoid.

Proof.

If preinverses are unique then \mathopen{}\mathclose{{}\left(xx^{-1}}\right) and \mathopen{}\mathclose{{}\left(x^{-1}x}\right) are uniquely determined by $x$, so canonical local units are unique. ∎

Corollary 8.3**.**

If $S$ is an inverse semigroup then S\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a groupoid.

It is clear that Corollaries 8.2 and 8.3 are related to the so-called Ehresmann-Schein-Nampooribad theorem in semigroup theory: groupoids correspond to inverse semigroups and, more generally, to regular semigroups whose canonical local units are unique.

8.3. From left skew-prepoloids

to left skew-poloids

Definition 8.3**.**

Let $P$ be* a left skew-prepoloid with a binary operation \mathfrak{m}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(xy}\right), $x,y\in P$. The restricted multiplication* on the carrier set of $P$ is a binary operation \boldsymbol{\mathsf{m}}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(x\cdot y}\right) such that if \mathopen{}\mathclose{{}\left(x\cdot y}\right) and \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right), and \mathopen{}\mathclose{{}\left(x\cdot y}\right) if and only if there is some local left unit $\lambda_{y}$ for $y$ such that \mathopen{}\mathclose{{}\left(x\lambda_{y}}\right)=x.

Note that if \mathopen{}\mathclose{{}\left(x\cdot y}\right) then \mathopen{}\mathclose{{}\left(xy}\right) by Proposition 6.11, so then \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right).

Lemma 8.3**.**

Let $P$ be a left skew-prepoloid with unique local left units. If \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)=\lambda_{x}=\lambda{}_{\lambda_{x}} for all $x\in P$ and \lambda_{\mathopen{}\mathclose{{}\left(xy}\right)}=\lambda_{x} for all $x,y\in P$ such that \mathopen{}\mathclose{{}\left(xy}\right) then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a left skew-poloid where $\varphi_{x}=\lambda_{x}$ for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right].

Proof.

If \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right) then \mathopen{}\mathclose{{}\left(y\lambda_{z}}\right)=y, so x=\mathopen{}\mathclose{{}\left(x\lambda_{\mathopen{}\mathclose{{}\left(y\cdot z}\right)}}\right)=\mathopen{}\mathclose{{}\left(x\lambda_{\mathopen{}\mathclose{{}\left(yz}\right)}}\right)=\mathopen{}\mathclose{{}\left(x\lambda_{y}}\right), so \mathopen{}\mathclose{{}\left(x\cdot y}\right), so \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right). Thus, \mathopen{}\mathclose{{}\left(xy}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(y\lambda_{z}}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)\lambda_{z}}\right), so \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)\cdot z}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right). Hence, if \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right)}\right) then \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right).

If \mathopen{}\mathclose{{}\left(x\cdot y}\right) and \mathopen{}\mathclose{{}\left(y\cdot z}\right) then \mathopen{}\mathclose{{}\left(x\lambda_{y}}\right)=x and \mathopen{}\mathclose{{}\left(y\lambda_{z}}\right)=y. Hence, x=\mathopen{}\mathclose{{}\left(x\lambda_{y}}\right)=\mathopen{}\mathclose{{}\left(x\lambda_{\mathopen{}\mathclose{{}\left(yz}\right)}}\right)=\mathopen{}\mathclose{{}\left(x\lambda_{\mathopen{}\mathclose{{}\left(y\cdot z}\right)}}\right), so \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right), so again \mathopen{}\mathclose{{}\left(x\cdot\mathopen{}\mathclose{{}\left(y\cdot z}\right)}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(yz}\right)}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy}\right)z}\right)=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot y}\right)\cdot z}\right).

If \mathopen{}\mathclose{{}\left(y\cdot\lambda_{x}}\right) then \mathopen{}\mathclose{{}\left(y\lambda_{\lambda_{x}}}\right)=y so \mathopen{}\mathclose{{}\left(y\cdot\lambda_{x}}\right)=\mathopen{}\mathclose{{}\left(y\lambda_{x}}\right)=y since $\lambda_{x}=\lambda_{\lambda_{x}}$. We also have \mathopen{}\mathclose{{}\left(\lambda_{x}\lambda_{x}}\right)=\lambda_{x}, so \mathopen{}\mathclose{{}\left(\lambda_{x}\cdot x}\right), so \mathopen{}\mathclose{{}\left(\lambda_{x}\cdot x}\right)=\mathopen{}\mathclose{{}\left(\lambda_{x}x}\right)=x. Thus, $\lambda_{x}$ is a right unit and a local left unit for $x$, that is, a twisted left unit $\varphi_{x}$ for $x$. The uniqueness of $\varphi_{x}$ follows from the uniqueness of $\lambda_{x}$. ∎

Combining Lemma 8.3 with Propositions 6.12 and 6.13, we obtain the following results:

Theorem 8.4**.**

If $P$ is a left skew-prepoloid with unique local left units then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a left skew-poloid.

Corollary 8.4**.**

If $S$ is a left unital semigroup with unique local left units then S\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a left skew-poloid.

The fact that a left skew-poloid can be constructed from a left skew-prepoloid is related to the fact that an inductive constellation can be constructed from a left restriction semigroup [6, 4].

8.4. From left skew-pregroupoids

to left skew-groupoids

Definition 8.4**.**

Let $P$ be* a left skew-pregroupoid with binary operation \mathfrak{m}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(xy}\right), $x,y\in P$. The restricted binary operation* on the carrier set of $P$ is a binary operation \boldsymbol{\mathsf{m}}:\mathopen{}\mathclose{{}\left(x,y}\right)\mapsto\mathopen{}\mathclose{{}\left(x\cdot y}\right) such that if \mathopen{}\mathclose{{}\left(x\cdot y}\right) and \mathopen{}\mathclose{{}\left(xy}\right) then \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right), and \mathopen{}\mathclose{{}\left(x\cdot y}\right) if and only if there is a canonical local left unit \boldsymbol{\uplambda}_{y}=\mathopen{}\mathclose{{}\left(yy^{-1}}\right) such that \mathopen{}\mathclose{{}\left(x\boldsymbol{\uplambda}_{y}}\right)=x .

If \mathopen{}\mathclose{{}\left(x\cdot y}\right) then \mathopen{}\mathclose{{}\left(x\boldsymbol{\uplambda}_{y}}\right) so \mathopen{}\mathclose{{}\left(xy}\right) by Proposition 6.11, so \mathopen{}\mathclose{{}\left(x\cdot y}\right)=\mathopen{}\mathclose{{}\left(xy}\right).

Lemma 8.4**.**

Let $P$ be a left skew-pregroupoid with unique canonical local left units. If \mathopen{}\mathclose{{}\left(\boldsymbol{\uplambda}_{x}\boldsymbol{\uplambda}_{x}}\right)=\boldsymbol{\uplambda}_{x}=\boldsymbol{\uplambda}_{\boldsymbol{\uplambda}_{x}} for all $x\in P$ and \boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(xy}\right)}=\boldsymbol{\uplambda}_{x} for all $x,y\in P$ such that \mathopen{}\mathclose{{}\left(xy}\right) then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a left skew-groupoid where $\varphi_{x}=\boldsymbol{\uplambda}_{x}$ and $\varphi_{x^{-1}}=\boldsymbol{\uplambda}_{x^{-1}}$ for all x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right].

Proof.

It can be proved that $P$ is a left skew-poloid with unique twisted left units $\varphi_{x}=\boldsymbol{\uplambda}_{x}$ by using the argument in the proof of Lemma 8.3 again. To complete the proof, we first show that for every x\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] there is a corresponding right preinverse x^{-1}\in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right].

If $x^{-1}\in P$ is a right preinverse of $x$ then \mathopen{}\mathclose{{}\left(xx^{-1}}\right)=\boldsymbol{\uplambda}_{x} and \mathopen{}\mathclose{{}\left(x^{-1}x}\right)=\boldsymbol{\uplambda}_{x^{-1}}. Thus,

[TABLE]

so \mathopen{}\mathclose{{}\left(x\cdot x^{-1}}\right) and \mathopen{}\mathclose{{}\left(x^{-1}\cdot x}\right) since \boldsymbol{\uplambda}_{x^{-1}}\!=\!\boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)} by the uniqueness of canonical local left units (Section 4.4). Hence, \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x\cdot x^{-1}}\right)\cdot x}\right) and \mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(x^{-1}\cdot x}\right)\cdot x^{-1}}\right), so

[TABLE]

so $x^{-1}$ is a right preinverse of $x$ in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right].

It is shown in the proof of Lemma 8.3 that if \mathopen{}\mathclose{{}\left(y\cdot\lambda_{x}}\right) then \mathopen{}\mathclose{{}\left(y\cdot\lambda_{x}}\right)=y for any $x,y\in P$. It can be shown by a similar argument that if \mathopen{}\mathclose{{}\left(y\cdot\boldsymbol{\uplambda}_{x}}\right) then \mathopen{}\mathclose{{}\left(y\cdot\boldsymbol{\uplambda}_{x}}\right)=y, and that if \mathopen{}\mathclose{{}\left(y\cdot\boldsymbol{\uplambda}_{x^{-1}}}\right) then \mathopen{}\mathclose{{}\left(y\cdot\boldsymbol{\uplambda}_{x^{-1}}}\right)=\mathopen{}\mathclose{{}\left(y\cdot\boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)}}\right)=y. Hence, $x^{-1}$ is a strong right preinverse of $x$ in P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] with $\varphi_{x}=\boldsymbol{\uplambda}_{x}$ and \varphi_{x^{-1}}=\boldsymbol{\uplambda}_{x^{-1}}=\boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)}.

The uniqueness of $\varphi_{x^{-1}}$ follows from the uniqueness of \boldsymbol{\uplambda}_{\mathopen{}\mathclose{{}\left(x^{-1}}\right)}. ∎

Combining Lemma 8.4 with Propositions 6.16 and 6.17, we obtain the following results:

Theorem 8.5**.**

If $P$ is a left skew-pregroupoid with unique canonical local left units, then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a left skew-groupoid.

Theorem 8.6**.**

If $P$ is a left skew-pregroupoid with unique preinverses then P\mathopen{}\mathclose{{}\left[\boldsymbol{\mathsf{m}}}\right] is a left skew-groupoid.

Proof.

If preinverses are unique then \mathopen{}\mathclose{{}\left(xx^{-1}}\right) is determined by $x$, so canonical local left units are unique. ∎

8.5. Final remarks

Sections 6 and 7 are structured around three main distinctions, exemplified by the distinctions between poloids and skew-poloids, poloids and prepoloids, and poloids and groupoids. These distinctions generate eight types of poloid-like magmoids: prepoloids, pregroupoids, skew-prepoloids, skew-pregroupoids, poloids, groupoids, skew-poloids, and skew-groupoids.

The most superficial of the three main distinctions is perhaps the one between poloid-like and skew-poloid-like magmoids. The equivalence of these two notions was briefly discussed in Section 7.2. The idea that the difference between them ultimately reflects the way mappings are formalized – as functions with domains and codomains, corresponding to the two-sided objects, or as prefunctions with only domains, corresponding to the one-sided objects – is implicit in [9].101010In particular, a transformation magmoid is associative, whereas a pretransformation magmoid is just skew-associative; see Facts 3 and 4 in [9]. The close connection between the two notions does not mean, though, that the corresponding distinction is trivial or of little interest; showing the material equivalence of superficially different notions is an important accomplishment in mathematics. It seems that much work remains to be done in this connection.

Next, the distinction between poloid-like and prepoloid-like magmoids is also a distinction between two closely related notions. We have seen in Section 8 that (skew-)prepoloids with unique local units and (skew-)pregroupoids with unique preinverses can be transformed into corresponding (skew-)poloids and (skew-)groupoids. Conversely, it is shown in the literature that prepoloid-like magmoids can be recovered from corresponding poloid-like magmoids with additional structure. The relationship between prepoloid-like and poloid-like magmoids, in particular semigroups and poloids/groupoids, has been researched extensively, but not exhaustively, starting with Ehresmann.

Finally, there is the distinction between different kinds of poloid-like magmoids exemplified by that between monoids and groups. This is obviously a significant distinction, though one which hardly needs further comment here.

Appendix A Heap-like algebras with partial operations

As we know, a group, with a binary operation \mathopen{}\mathclose{{}\left(x,y}\right)\mapsto xy, is closely related to a corresponding heap with a ternary operation, written \mathopen{}\mathclose{{}\left(x,y,z}\right)\mapsto\mathopen{}\mathclose{{}\left[x,y,z}\right] or \mathopen{}\mathclose{{}\left(x,y,z}\right)\mapsto\mathopen{}\mathclose{{}\left[xyz}\right]. Analogously, an involution magmoid, with a partial binary operation \mathopen{}\mathclose{{}\left(x,y}\right)\mapsto xy and total unary operation $x\mapsto x^{*}$, is closely related to a corresponding algebra with a partial ternary operation \mathopen{}\mathclose{{}\left(x,y,z}\right)\mapsto\mathopen{}\mathclose{{}\left[xyz}\right] as described below.

Semiheapoids and semiheaps from involution semigroupoids

One can use any total involution $*$ on a semigroupoid $P$ to define a partial ternary operation

[TABLE]

on $P$, with \mathopen{}\mathclose{{}\left[xyz}\right] being defined if and only if \mathopen{}\mathclose{{}\left(xy^{*}z}\right).

Let \mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left[xyz}\right]uv}\right], \mathopen{}\mathclose{{}\left[xy\mathopen{}\mathclose{{}\left[zuv}\right]}\right] and \mathopen{}\mathclose{{}\left[x\mathopen{}\mathclose{{}\left[uzy}\right]v}\right] be defined. Then \mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left[xyz}\right]uv}\right]=\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xy^{*}z}\right)u^{*}v}\right)=\mathopen{}\mathclose{{}\left(xy^{*}zu^{*}v}\right), \mathopen{}\mathclose{{}\left[xy\mathopen{}\mathclose{{}\left[zuv}\right]}\right]=\mathopen{}\mathclose{{}\left(xy^{*}\mathopen{}\mathclose{{}\left(zu^{*}v}\right)}\right)=\mathopen{}\mathclose{{}\left(xy^{*}zu^{*}v}\right) and \mathopen{}\mathclose{{}\left[x\mathopen{}\mathclose{{}\left[uzy}\right]v}\right]=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(uz^{*}y}\right)^{*}v}\right)=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(y^{*}\mathopen{}\mathclose{{}\left(z^{*}}\right)^{*}u^{*}}\right)v}\right)=\mathopen{}\mathclose{{}\left(xy^{*}zu^{*}v}\right). Thus,

[TABLE]

Extending Wagner’s terminology [18], we call any set $P$ with a partial ternary operation \mathsf{t}:\mathopen{}\mathclose{{}\left(x,y,z}\right)\mapsto\mathopen{}\mathclose{{}\left[xyz}\right] satisfying (A.1) when all terms are defined a semiheapoid. If $\mathsf{t}$ is a total function satisfying (A.1) then $P$ is a semiheap.

Semiheapoids and semiheaps from groupoids; heapoids and heaps

By Propositions 7.9 and 7.10, every groupoid has a total involution, namely the function $x\mapsto x^{-1}$. In this case, the identities (7.2) give rise to identities applying to $\mathsf{t}$. Specifically, if \mathopen{}\mathclose{{}\left(xx^{-1}y}\right) then \mathopen{}\mathclose{{}\left(xx^{-1}y}\right)=\mathopen{}\mathclose{{}\left(\ell_{x}y}\right)=y and if \mathopen{}\mathclose{{}\left(yx^{-1}x}\right) then \mathopen{}\mathclose{{}\left(yx^{-1}x}\right)=\mathopen{}\mathclose{{}\left(yr_{x}}\right)=y, so

[TABLE]

We call a non-empty set $P$ with a partial ternary operation $\mathsf{t}$ satisfying (A.1) and (A.2) when all terms are defined a heapoid. If $\mathsf{t}$ is a total function satisfying (A.1) and (A.2) then $P$ is a heap.

Semiheapoids and semiheaps from pregroupoids; preheapoids and preheaps

By Propositions 6.7 and 6.8, every pregroupoid $P$ with unique preinverses has a total involution, namely the function $x\mapsto x^{-1}$. If the involution has this form then

[TABLE]

since idempotents in pregroupoids with unique preinverses commute, so

[TABLE]

We also have \mathopen{}\mathclose{{}\left(xx^{-1}x}\right)=x, so

[TABLE]

We call a non-empty set $P$ with a total function $\mathsf{t}$ satisfying (A.1), (A.3) and (A.4) a preheap or, in Wagner’s terminology, a generalized heap; if $\mathsf{t}$ is a partial function satisfying (A.1), (A.3) and (A.4) when all terms are defined, we get a preheapoid instead.

Generalized semiheapoids

We can let not only the binary operation on a semigroupoid $P$ but also the involution $*$ on $P$ be a partial function. Then \mathopen{}\mathclose{{}\left[xyz}\right] is defined if and only if \mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(y^{*}}\right)z}\right), and then \mathopen{}\mathclose{{}\left[xyz}\right]=\mathopen{}\mathclose{{}\left(x\mathopen{}\mathclose{{}\left(y^{*}}\right)z}\right). Such generalized semiheapoids can be defined naturally for semigroupoids of matrices (see Section 5.2).