On the implication $T_{0} \Rightarrow T_{3 \frac{1}{2}}$ for some   topological protomodular algebras

Dali Zangurashvili

arXiv:1905.07600·math.CT·May 21, 2019

On the implication $T_{0} \Rightarrow T_{3 \frac{1}{2}}$ for some topological protomodular algebras

Dali Zangurashvili

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TL;DR

This paper introduces right-cancellable topological protomodular algebras and proves that such algebras satisfying the T0 separation axiom are necessarily completely regular, linking algebraic properties with topological regularity.

Contribution

It establishes a new connection between algebraic cancellability and topological regularity in protomodular algebras, expanding understanding of their structure.

Findings

01

Right-cancellable topological protomodular algebras are completely regular under T0 separation.

02

The notion of right-cancellable protomodular algebra is introduced.

03

A key implication T0 ⇒ T3 1/2 is demonstrated for these algebras.

Abstract

The notion of a right-cancellable protomodular algebra is introduced. It is proved that a right-cancellable topological protomodular algebra that satisfies the separation axiom $T_{0}$ is completely regular.

Equations162

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)) =

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)) =

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'})),

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'})),

α_{i} (a, a) = e_{i};

α_{i} (a, a) = e_{i};

θ (α_{1} (a, b), α_{2} (a, b), ..., α_{n} (a, b), b) = a .

θ (α_{1} (a, b), α_{2} (a, b), ..., α_{n} (a, b), b) = a .

θ (a, b) = ab,

θ (a, b) = ab,

α (a, b) = a / b,

α (a, b) = a / b,

θ (a_{1}, a_{2}, ..., a_{n}, θ (b_{1}, b_{2}, ..., b_{n}, c)) =

θ (a_{1}, a_{2}, ..., a_{n}, θ (b_{1}, b_{2}, ..., b_{n}, c)) =

= θ (θ (a_{1}, a_{2}, ..., a_{n}, b_{1}), θ (a_{1}, a_{2}, ..., a_{n}, b_{2}), ..., θ (a_{1}, a_{2}, ..., a_{n}, b_{n}), c) .

= θ (θ (a_{1}, a_{2}, ..., a_{n}, b_{1}), θ (a_{1}, a_{2}, ..., a_{n}, b_{2}), ..., θ (a_{1}, a_{2}, ..., a_{n}, b_{n}), c) .

i = 1 ⋂ n α_{i} (-, a)^{- 1} (H_{i}),

i = 1 ⋂ n α_{i} (-, a)^{- 1} (H_{i}),

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)) =

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)) =

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'}));

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'}));

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)), θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b)) =

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)), θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b)) =

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'})), θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}));

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'})), θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}));

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b)) =

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b)) =

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'})));

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'})));

θ (a_{1}, a_{2}, ..., a_{n}, b) = θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'}),

θ (a_{1}, a_{2}, ..., a_{n}, b) = θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'}),

α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}), b) = T_{i} (a_{1}, a_{2}, ..., a_{n}, a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''});

α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}), b) = T_{i} (a_{1}, a_{2}, ..., a_{n}, a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''});

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), b) =

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b), b) =

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), b^{'})

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, b^{'}), b^{'})

α_{i} (b, θ (a_{1}, a_{2}, ..., a_{n}, b)) =

α_{i} (b, θ (a_{1}, a_{2}, ..., a_{n}, b)) =

= α_{i} (b^{'}, θ (a_{1}, a_{2}, ..., a_{n}, b^{'})) .

= α_{i} (b^{'}, θ (a_{1}, a_{2}, ..., a_{n}, b^{'})) .

c_{i} = α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)) .

c_{i} = α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)) .

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)), θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b)) =

α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)), θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b)) =

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)), θ (c_{1}, c_{2}, ..., c_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b))) .

= α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b)), θ (c_{1}, c_{2}, ..., c_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b))) .

θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}) = θ (t_{1}, t_{2}, ..., t_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'})),

θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}) = θ (t_{1}, t_{2}, ..., t_{n}, θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, b^{'})),

t_{i} = α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, e_{i}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, e_{i})) .

t_{i} = α_{i} (θ (a_{1}, a_{2}, ..., a_{n}, e_{i}), θ (a_{1}^{'}, a_{2}^{'}, ..., a_{n}^{'}, e_{i})) .

θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}) = θ (t_{1}, t_{2}, ..., t_{n}, θ (a_{1}, a_{2}, ..., a_{n}, b)) .

θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}) = θ (t_{1}, t_{2}, ..., t_{n}, θ (a_{1}, a_{2}, ..., a_{n}, b)) .

α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}), b) = α_{i} (θ (t_{1}, t_{2}, ..., t_{n}, θ (a_{1}, a_{2}, ..., a_{n}, b)), b) .

α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}), b) = α_{i} (θ (t_{1}, t_{2}, ..., t_{n}, θ (a_{1}, a_{2}, ..., a_{n}, b)), b) .

α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}), b) = T_{i} (a_{1}, a_{2}, ..., a_{n}, t_{1}, t_{2}, ..., t_{n})

α_{i} (θ (a_{1}^{''}, a_{2}^{''}, ..., a_{n}^{''}, b^{'}), b) = T_{i} (a_{1}, a_{2}, ..., a_{n}, t_{1}, t_{2}, ..., t_{n})

t_{i} = α_{i} (b, θ (a_{1}, a_{2}, ..., a_{n}, b)) .

t_{i} = α_{i} (b, θ (a_{1}, a_{2}, ..., a_{n}, b)) .

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TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Advanced Algebra and Logic

Full text

On the implication $T_{0}\Rightarrow T_{3\frac{1}{2}}$ for some

topological protomodular algebras

Dali Zangurashvili

Abstract.

The notion of a right-cancellable protomodular algebra is introduced. It is proved that a right-cancellable topological protomodular algebra that satisfies the separation axiom $T_{0}$ is completely regular.

Key words and phrases: topological protomodular algebra; separation axioms; completely regular space; uniformity.

2000 Mathematics Subject Classification: 22A30, 22A26, 08B05, 18C10, 54H12.

1. Introduction

One of the classical results of topological group theory asserts that any $T_{0}$ group is completely regular (and hence Hausdorff). There naturally arises the question whether this statement remains valid if we replace ”group” by an algebra from any variety of universal algebras. A part of this assertion related to the Hausdorff condition (i.e., that $T_{0}$ implies $T_{2}$ ) was studied first by W. Taylor [13], who proved that this is the case if a variety is congruence permutable. Subsequently, this result was step by step improved in the papers [9],[5],[6],[12]. To the best of our knowledge, the strongest result obtained in this direction asserts that any $T_{0}$ algebra is Hausdorff if a variety is $k$ -permutable and congruence modular (K. Kearnes, L. Sequeira [12]).

The topological algebras for another class of varieties were considered by P. T. Johnstone and M. C. Pedicchio [10]. Namely, they studied the category of topological Mal’cev algebras, and, in particular, proved the implication $T_{0}\Rightarrow T_{2}$ for them. They also proved that if, in addition, Mal’cev term on an algebra is weakly associative in the sense of [10], then this algebra is regular. The issue whether the separation axiom $T_{1}$ implies regularity was further studied by F. Borceux and M. M. Clementino in [1] and [2], where they generalized many properties of topological groups to the case of protomodular algebras, and, in particular, proved that any $T_{1}$ protomodular algebra is regular. Note that the notion of a protomodular variety is relatively recent and derived from the notion of a protomodular category introduced by D. Bourn as an abstract setting in which many properties of groups remain valid [3]. Note further that there is the purely syntactical characterization of a protomodular variety due to D. Bourn and G. Janelidze [4]. It requires the existence of one operation $\theta$ (called a protomodular/semi-abelian operation) of arbitrarily high arity $(n+1)$ , together with some binary operations $\alpha_{i}$ and constants $e_{i}$ that satisfy certain identities.

The aim of this paper is to establish for which protomodular algebras the separation axiom $T_{0}$ implies complete regularity. One relevant sufficient condition is immediate: this is the case if the algebraic theory of a variety has a group reduct. For this, there are two criteria: one criterion requires the existence of an associative Mal’cev operation (Johnstone-Pedicchio [10]), and the other requires the existence of a 2-associative semi-abelian operation [15].

Another approach to the problem in question is to attempt to generalize the well-known result, asserting that any topology on a group that satisfies the separation axiom $T_{0}$ is determined by a uniformity, to the case where ”group” is replaced by an algebra from a protomodular variety. Following this approach we distinguish a certain subvariety. It is determined by the identities

[TABLE]

$1\leq i\leq n$ . We call the algebras satisfying these identities right-cancellable. We prove:

A right-cancellable topological protomodular algebra that satisfies the separation axiom $T_{0}$ is completely regular.

The results of this paper were announced in [14] without proofs.

2. Preliminaries

For the definition of a protomodular category we refer the reader to the paper [3] by D. Bourn.

Let $\mathbb{V}$ be a variety of universal algebras of a type $\mathcal{F}$ .

Theorem 2.1.

(Bourn-Janelidze [4]) $\mathbb{V}$ is protomodular if and only if its algebraic theory contains, for some natural $n$ , constants $e_{1},e_{2},...,e_{n}$ , binary operations $\alpha_{1}$ , $\alpha_{2}$ ,…, $\alpha_{n}$ and an $(n+1)$ -ary operation $\theta$ such that the following identities are satisfied:

[TABLE]

For simplicity, algebras from a protomodular variety are called protomodular. The operation $\theta$ satisfying (2.2) for some $\alpha_{i}$ and $e_{i}$ which in their turn satisfy (2.1) is called protomodular. A protomodular operation is called semi-abelian if all $e_{i}$ ’s are equal to one another.

The motivating example of a protomodular variety is given by the variety of groups. More generally, any variety whose algebraic theory contains a group operation is protomodular (for this reason the variety of Boolean algebras is protomodular). In that case we have:

[TABLE]

and $e$ is the unit of the group. The other examples of protomodular varieties are given by the varieties of left/right semi-loops, loops, locally Boolean distributive lattices [1], Heyting algebras, Heyting semi-lattices [11]. Observe that the operations (2.3) and (2.4) serve as operations from the Bourn-Janelidze criterion in the case of left semi-loops and of loops too.

The identities (2.1) and (2.2) immediately imply [2]:

(a) if $\alpha_{i}(a,c)=\alpha_{i}(b,c)$ , for all $i$ ( $1\leq i\leq n$ ), then $a=b$ ;

(b) if $\alpha_{i}(a,b)=e_{i}$ , for all $i$ ( $1\leq i\leq n$ ), then $a=b$ ;

(c) $\theta(e_{1},e_{2},...,e_{n},a)=a.$

Let $A$ be a set and let $\theta$ be an $(n+1)$ -ary operation on $A$ . The two notions of associativity for $\theta$ are introduced in [15]. 1-Associativity is the straightforward generalization of the usual associativity condition given by the move of parentheses. $\theta$ is called 2-associative [15] if, for any $a_{1},a_{2},...,a_{n},b_{1},b_{2},...,b_{n},c\in A$ , one has

[TABLE]

Throughout the paper $\mathbb{V}_{n}$ denotes the simplest protomodular variety, i.e. the variety with the signature $\mathfrak{F}_{n}$ containing only one $(n+1)$ -ary operation symbol $\theta$ , the binary operation symbols $\alpha_{1},\alpha_{2},...,\alpha_{n}$ , and the constant symbols $e_{1},e_{2},...,e_{n}$ , where the identities are (2.1) and (2.2).

Let now $\mathbb{V}$ be any variety of universal algebras, and let $A$ be an algebra from $\mathbb{V}$ . $A$ is called a topological algebra if $A$ is equipped with a topology such that all operations from $\mathcal{F}$ are continuous.

Proposition 2.2.

(Borceux-Clementino [2]). Let $a$ be an element of a topological protomodular algebra $A$ . Then the subsets

[TABLE]

with $H_{i}$ being an open neighborhood of $e_{i}$ , constitute a base of neighborhoods of $a$ .

We refer the reader to [7] for the needed definitions and facts from the uniform spaces theory. Here we only recall that a uniformity on a set $X$ is a family of binary relations on $X$ , that satisfies certain conditions. Any uniformity $\mathrm{U}$ determines a topology on $X$ as follows: $O$ is open if and only if, for any $x\in O$ , there exists $R\in\mathrm{U}$ such that $B(x,R)\subset O$ ; here $B(x,R)$ denotes the set $\{y|(x,y)\in R\}$ . Moreover, if a family $\mathbf{C}$ of coverings of a set $X$ satisfies the conditions (C1)-(C4) below, then the family of sets $(\cup_{H\in\mathcal{A}}H\times H)_{\mathcal{A}\in\mathbf{C}}$ is a base of a uniformity on $X$ .

(C1) If $\mathcal{A}\in\mathbf{C}$ and $\mathcal{A}$ is inscribed in a covering $\mathcal{B}$ (i.e., for any $A\in\mathcal{A}$ there exists $B$ from $\mathcal{B}$ , with $A\subset B$ ), then $\mathcal{B}\in\mathbf{C}$ ;

(C2) for any $\mathcal{A}_{1},\mathcal{A}_{2}\in\mathbf{C}$ , there is $\mathcal{A}\in\mathbf{C}$ , which is inscribed in both $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ ;

(C3) for any $\mathcal{A}\in\mathbf{C}$ , there exists $\mathcal{B}\in\mathbf{C}$ which is strongly star-like inscribed in $\mathcal{A}$ (i.e., for any $B\in\mathcal{B}$ there exists $A$ from $\mathcal{A}$ , with $St(B,\mathcal{B})\subset A$ );

(C4) for any distinct $x,y\in X$ , there exists $\mathcal{A}\in\mathbf{C}$ such that, for any $A\in\mathcal{A}$ we have $\{x,y\}\nsubseteq\mathcal{A}$ .

For a set $M$ of $X$ and a covering $\mathcal{A}$ of $X$ , the symbol $St(M,\mathcal{A})$ denotes the set $\bigcup_{A\in\mathcal{A},A\bigcap M\neq\emptyset}A$ .

3. Right-Cancellable Protomodular Algebras

Lemma 3.1.

Let $\mathbb{V}$ be a protomodular variety, and let $A$ be a $\mathbb{V}$ -algebra. The conditions (i)-(iv) given below are equivalent and imply the condition (v). If $A$ is associative in any sense of [15], then all these conditions are equivalent.

(i)* For any $a_{1},a_{2},...,a_{n},a^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{n},b,b^{\prime}\in A$ and $i$ ( $1\leq i\leq n)$ , we have*

[TABLE]

(ii)* for any $a_{1},a_{2},...,a_{n},a^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{n},b,b^{\prime}\in A$ and $i$ ( $1\leq i\leq n)$ , we have*

[TABLE]

(iii)* for any $a_{1},a_{2},...,a_{n},a^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{n},b,b^{\prime}\in A$ and $i$ ( $1\leq i\leq n)$ , we have*

[TABLE]

(iv)* for any $i$ ( $1\leq i\leq n)$ , there is a term $T_{i}$ of $3n$ variables over the signature of $\mathbb{V}$ , such that for any $a_{1},a_{2},...,a_{n},a^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{n},a^{\prime\prime}_{1},a^{\prime\prime}_{2},...,a^{\prime\prime}_{n}$ , $b,b^{\prime}\in A,$ if*

[TABLE]

then

[TABLE]

(v)* for any $a_{1},a_{2},...,a_{n},b,b^{\prime}\in A$ , we have*

[TABLE]

and

[TABLE]

Proof.

(i) $\Rightarrow$ (ii): Let

[TABLE]

By (3.1) the value of $c_{i}$ does not depend on $b$ . From (2.2) we obtain

[TABLE]

Again applying (3.1) we obtain the desired result. One can prove similarly the implication (i) $\Rightarrow$ (iii).

(ii) $\Rightarrow$ (iv): First note that from (c) of Section 2 we conclude that (ii) implies (i). Let us now assume that (3.4) is satisfied. From (2.2) and (3.1) we have

[TABLE]

where

[TABLE]

Applying (3.4) we obtain

[TABLE]

Hence

[TABLE]

Then from (3.2) we have

[TABLE]

for the term $T_{i}=\alpha_{i}(\theta(t_{1},t_{2},...,t_{n},\theta(a_{1},a_{2},...,a_{n},e_{i})),e_{i})$ .

(iii) $\Rightarrow$ (iv): From (c) of Section 2 we obtain (3.7). Let (3.4) be satisfied, and let

[TABLE]

From (3.7) we have

[TABLE]

On the other hand, (3.4) and (3.7) imply that

[TABLE]

and hence, from (2.2) we obtain

[TABLE]

Then, taking into account (3.3), we obtain

[TABLE]

for the term $T_{i}=\alpha_{i}(\theta(a^{\prime\prime}_{1},a^{\prime\prime}_{2},...,a^{\prime\prime}_{n},e_{i}),\theta(t_{1},t_{2},...,t_{n},\theta(a^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{n},e_{i}))).$

(iv) $\Rightarrow$ (i): From (c) of Section 2 we have

[TABLE]

It implies (3.1).

The implication (i) $\Rightarrow$ (v) follows from (c) of Section 2.

(v) $\Rightarrow$ (i): Let $A$ be associative, and let

[TABLE]

By (3.7) the value of $c_{i}$ does not depend on $b$ . From (2.2) we have

[TABLE]

Applying the associativity and then (3.6) we obtain the required equality. ∎

Definition 3.2.

We call a protomodular algebra right-cancellable if it satisfies the equivalent conditions of Lemma 3.1.

* Remark 3.3**.*

(a) Let $n=1$ . Below we use the traditional abbreviation $ab$ for $\theta(a,b)$ , and $a/b$ for $\alpha(a,b)$ .

Let $A$ be an algebra from this variety, and the following equivalent conditions be satisfied***The implication (3.9) $\Rightarrow$ (3.8) is obvious. For the converse, observe that from (2.2) we have $(a/e)e=a$ ; on the other hand, (3.8) implies $(a/e)e=a/e.$ The conditions (3.8) and (3.9) are satisfied if, for instance, $A$ is commutative.:

[TABLE]

for any $a\in A$ . Then the condition $(i)$ implies the associativity. Indeed, from (3.8) we obtain the identity

[TABLE]

It implies

[TABLE]

for any $a,b,c\in A$ . Multiplying both parts of this equality by $(bc)$ from the right, we obtain the associativity condition. Then, taking into account the fact that any set equipped with an associative binary operation, which has a left identity and left inverses, is a group (see, for instance, [8]), we obtain that $A$ is a group.

Note that (3.1) implies (3.8) and (3.9) in the case of the variety of left semi-loops. Indeed, we have

[TABLE]

Thus we can conclude that any right-cancellable left semi-loop (loop) is a group.

(b) One can show that non-trivial Boolean algebras, locally Boolean distributive lattices, Heyting algebras, Heyting semi-lattices with the protomodular operations given in [2] and [11] are not right-cancellable.

The examples of non-trivial right-cancellable algebras are given in the next section.

4. Right-Cancellable Topological Protomodular Algebras

Throughout this section, unless specified otherwise, we assume that $\mathbb{V}$ is a protomodular variety of universal algebras (of a type $\mathcal{F}$ ).

Lemma 4.1.

*For any topological $\mathbb{V}$ -algebra, the separation axiom $T_{0}$ implies $T_{1}$ . *

Proof.

Let $A$ be a $\mathbb{V}$ -algebra, and let $a,b\in A$ . Let there be a neighbourhood $U$ of $a$ that does not contain $b$ . Without loss of generality one can assume that $U$ is given by (2.3) for some base $(H_{j})_{1\leq j\leq n}$ of neighbourhoods of $(e_{j})_{1\leq j\leq n}$ . Since $b\overline{\in}U$ , there is $j$ such that

[TABLE]

Since $\alpha_{j}(b;-)$ is continious and $\alpha_{j}(b,b)=e_{j}$ , there is a neighbourhood $H$ of $b$ such that $\alpha_{j}(b,H)\subset H_{j}$ . This implies that $a\overline{\in}H$ . ∎

Theorem 4.2.

Let $A$ be a right-cancellable $\mathbb{V}$ -algebra. Then any topology on $A$ that satisfies the separation axiom $T_{0}$ is completely regular.

We will prove this statement in two steps. Let $A$ satisfy the conditions (i)-(iv) of Lemma 3.1.

Lemma 4.3.

Let $A$ be a $T_{0}$ $\mathbb{V}$ -algebra. Let $\mathcal{B}_{i}$ be any base of neighbourhoods of $e_{i}$ , and let $H_{i}\in{B_{i}}$ $(1\leq i\leq n)$ . Let $H=(H_{1},H_{2},...,H_{n})$ . Consider the covering

[TABLE]

of $A$ . Let $\mathbf{C}$ be the family of all coverings of $A$ in which the coverings $\mathcal{C}_{H}$ are inscribed. Then the family $\mathbf{C}$ determines a uniformity on $A$ .

Proof.

Let us show that $\mathcal{C}_{H}$ satisfies the conditions (C1)-(C4).

The validity of (C1) is obvious. The condition (C2) easily follows from the fact that for any $H^{\prime}_{i},H^{\prime\prime}_{i}\in\mathcal{B}_{i}$ there exists $H_{i}\in\mathcal{B}$ with $H_{i}\subset H^{\prime}_{i}\bigcap H^{\prime\prime}_{i}$ .

To prove that the condition (C3) is satisfied, it is sufficient to show that, for any $H=(H_{1},H_{2},...,H_{n})\in\mathcal{B}_{1}\times\mathcal{B}_{2}\times...\mathcal{B}_{n}$ , there exists $H^{\prime}=(H^{\prime}_{1},H^{\prime}_{2},...,H^{\prime}_{n})\in\mathcal{B}_{1}\times\mathcal{B}_{2}\times...\mathcal{B}_{n}$ such that

[TABLE]

for any $a\in A$ . From the implication (3.4) $\Rightarrow$ (3.5) it follows that

[TABLE]

for any $i(1\leq i\leq n)$ , and since the mappings $T_{i}:A^{3n}\rightarrow A$ are continuous, there exist $H_{11}^{i},H_{12}^{i},H_{13}^{i}\in\mathcal{B}_{1},H_{21}^{i},H_{22}^{i},H_{23}^{i}\in\mathcal{B}_{2},...,H_{n1}^{i},H_{n2}^{i},H_{n3}^{i}$ $\in\mathcal{B}_{n}$ with

[TABLE]

Let

[TABLE]

for any $j$ $(1\leq j\leq n)$ . If

[TABLE]

for some $a^{\prime}$ from $A$ , then there exists $b$ such that

[TABLE]

and

[TABLE]

for all $i$ $(1\leq i\leq n)$ . We have

[TABLE]

and

[TABLE]

Hence

[TABLE]

Let

[TABLE]

Then $\alpha_{i}(c,a^{\prime})=h^{\prime\prime}_{i}\in H^{\prime}_{i}$ for all $i$ $(1\leq i\leq n)$ . We have

[TABLE]

By the condition (iv) of Lemma 3.1, (4.4) implies that

[TABLE]

From (4.2) it follows that $\alpha_{i}(c,a)\in H_{i}$ , and hence (4.1) is satisfied.

Let us now consider different points $a,a^{\prime}\in A$ . By (b) of Section 2, there exists $i$ $(1\leq i\leq n)$ such that

[TABLE]

Since $A$ is a $T_{0}$ -space, by Lemma 4.1 there exists $H_{i}\in\mathcal{B}_{i}$ such that

[TABLE]

Let

[TABLE]

denote the left (or right)-hand part of (3.1). We obviously have

[TABLE]

Since $t_{i}:A^{2n}\rightarrow A$ is continuous, there are

[TABLE]

with

[TABLE]

Let

[TABLE]

for any $j$ $(1\leq j\leq n)$ , and let

[TABLE]

Let us show that none of the elements of $\mathcal{C}_{\widetilde{H}}$ contains both $a$ and $a^{\prime}$ . Indeed, suppose that

[TABLE]

Then

[TABLE]

and

[TABLE]

for all $j$ ’s. We have

[TABLE]

Similarly,

[TABLE]

Then, by the condition (i) of Lemma 3.1, we have

[TABLE]

From (4.6) and (4.7) it follows that

[TABLE]

but this contradicts (4.5). ∎

Lemma 4.4.

The topology $\tau^{\prime}$ induced by the uniformity described in Lemma 4.3 coincides with the topology $\tau$ of $A$ .

Proof.

First note that the family of all binary relations

[TABLE]

is the base of the uniformity determined by $\mathbf{C}$ (see Section 2). A subset $O$ of $A$ is open in the topology $\tau^{\prime}$ induced by this uniformity if and only if for any $a\in O$ there exist $H_{1}\in\mathcal{B}_{1}$ , $H_{2}\in\mathcal{B}_{2}$ ,…, $H_{n}\in\mathcal{B}_{n}$ such that

[TABLE]

for $H=(H_{1},H_{2},...,H_{n})$ . Obviously, $\bigcap_{i=1}^{n}\alpha_{i}^{-1}(-,a)(H_{i})$ is open in $\tau$ and is contained in $St(a,\mathcal{C}_{H})$ . Therefore any subset $O$ being open in $\tau^{\prime}$ is also open in $\tau$ . For the converse, consider a subset $O$ of $A$ which is open in $\tau$ . Let $a\in O$ . According to Proposition 2.1, there exist $H_{1}\in\mathcal{B}_{1}$ , $H_{2}\in\mathcal{B}_{2}$ ,…, $H_{n}\in\mathcal{B}_{n}$ such that

[TABLE]

But, as shown in the proof of Lemma 4.3, there exists $(H^{\prime}_{1},H^{\prime}_{2},...,H^{\prime}_{n})\in\mathcal{B}_{1}\times\mathcal{B}_{2}\times\mathcal{B}_{n}$ such that (4.1) holds. Since

[TABLE]

$O$ is open in $\tau^{\prime}$ .

∎

Lemma 4.3 and Lemma 4.4 obviously imply Theorem 4.2.

Example 4.5.

Let $A=\{0,1\}$ . Let us introduce the structure of a $\mathbb{V}_{2}$ -algebra on $A$ as follows. Let $\theta(i,j,k)=k$ if $i\neq j$ and $\theta(i,j,k)=1-k$ if $i=j$ . Moreover, let $\alpha_{1}(i,j)$ be [math], for any $i,j$ ; let $\alpha_{2}(i,j)$ be 0 if $i\neq j$ , and be 1 if $i=j$ . Besides, let $e_{1}=0$ and $e_{2}=1$ .

The condition (3.1) is obviously satisfied for $i=1$ . It is also satisfied for $i=2$ since the value of the left-hand side of (3.1) depends only on whether $a_{1}$ and $a^{\prime}_{1}$ are equal respectively to $a_{2}$ and $a^{\prime}_{2}$ . This implies that $A$ is a right-cancellable algebra.

This example in itself can not be considered as an example illustrating Theorem 4.2, since any topology on $A$ that satisfies $T_{1}$ is discrete. However, this example gives rise a lot of contensive ones. Indeed, for any set $I$ , and any congruence $R$ on $(\prod_{I}A)$ , the algebra $(\prod_{I}A)/R$ obviously also lies in the variety of right-cancellable $V_{2}$ -algebras. By Theorem 4.2, any topology on $(\prod_{I}A)/R$ that satisfies $T_{0}$ is completely regular.

5. Acknowledgment

This work is supported by Shota Rustaveli National Science Foundation of Georgia (Ref.: FR-18-10849).

Bibliography15

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] F. Borceux, M. M. Clementino, Topological protomodular algebras, Theory Appl. Categ. , 153(2006), 3085-3100.
2[2] F. Borceux, M. M. Clementino, Topological semi-Abelian algebras, Adv. Math , 190(2005), 425-453.
3[3] D. Bourn, Normalization equivalence, kernel equivalence and affine categories, Lecture notes in Math , 1448(1991), 43-62.
4[4] D. Bourn, G. Janelidze, Characterization of protomodular varieties of universal algebra, Theory Appl. Categ. , 11(2002), 143-147.
5[5] J. P. Coleman, Separation in topological algebras, Algebra Universalis , 35(1996), 72-84.
6[6] J. P. Coleman, Topological equivalents to n 𝑛 n -permutability, Algebra Universalis , 38(1997), 200-209.
7[7] R. Engelking, General topology, W.: Polish Scientific Publishers, 1977.
8[8] J. B. Fraleigh, A first course in Abstract Algebra, Person, 2004.

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On the implication T0⇒T312T_{0}\Rightarrow T_{3\frac{1}{2}}T0​⇒T321​​ for some

Abstract.

1. Introduction

2. Preliminaries

** Theorem 2.1****.**

** Proposition 2.2****.**

3. Right-Cancellable Protomodular Algebras

** Lemma 3.1****.**

Proof.

** Definition 3.2****.**

* Remark 3.3**.*

4. Right-Cancellable Topological Protomodular Algebras

** Lemma 4.1****.**

Proof.

** Theorem 4.2****.**

** Lemma 4.3****.**

Proof.

** Lemma 4.4****.**

Proof.

** Example 4.5****.**

5. Acknowledgment

On the implication $T_{0}\Rightarrow T_{3\frac{1}{2}}$ for some

Theorem 2.1.

Proposition 2.2.

Lemma 3.1.

Definition 3.2.

Lemma 4.1.

Theorem 4.2.

Lemma 4.3.

Lemma 4.4.

Example 4.5.