Supertropical Monoids II: Lifts, Transmissions, and Equalizers

TL;DR

This paper advances the theory of supertropical monoids and semirings by exploring lifts, transmissions, and equivalence relations, facilitating tangible factorizations and deepening understanding of their algebraic structure.

Contribution

It introduces explicit classifications of equivalence relations on supertropical monoids and analyzes their properties, addressing challenges from ghost products of tangible elements.

Findings

Constructed and classified equivalence relations on supertropical monoids.

Analyzed properties of these relations with focus on ghost product issues.

Enhanced understanding of factorizations in supertropical algebra.

Abstract

The category $\operatorname{STROP}$ of commutative semirings, whose morphisms are transmissions, is a full and reflective subcategory of the category $\operatorname{STROP}_m$ of supertropical monoids. Equivalence relations on supertropical monoids are constructed easily, and utilized effectively for supertropical semirings, whereas ideals are too special for semirings. Aiming for tangible factorizations, certain types of such equivalence relations are constructed and classified explicitly in this paper, followed by a profound study of their characteristic properties with special emphasis on difficulties arising from ghost products of tangible elements.

Equations

$$G(\psi ):=\psi (R)\cap M,$$

$$\alpha ={\alpha }_m\circ {\alpha }_t,$$

$$a\le b\iff a+b=b.$$

$$\forall x\in U:ex=0\Rightarrow x=0.$$

$$\rho :={\nu }_U|N:N⟶M$$

$$U=M\cup N,\text{where }M\cap N=\{0\}.$$

$$x\cdot y=\left\{\begin{array}[]{lll}x\cdot_{N}y&\text{if}&x,y\in N,\\[2.84526pt] \rho(x)\cdot_{M}y&\text{if}&x\in N,y\in M,\\[2.84526pt] x\cdot_{M}\rho(y)&\text{if}&x\in M,y\in N,\\[2.84526pt] x\cdot_{M}y&\text{if}&x,y\in M.\\ \end{array}\right.$$

$$\mathrm{STR}(\lambda ,\mu ):U^{\prime }=N^{\prime }\cup M^{\prime }⟶U=N\cup M,$$

$$U^0(v)=\mathrm{STR}(R/E,M,\bar{v}).$$

$$\tau^{-1}(x)=\left\{\begin{array}[]{lll}\{x,\tilde{x}\}&\text{if}&x\in M\cap N,\\[2.84526pt] \{\tilde{x}\}&\text{if}&x\in N\setminus M,\\[2.84526pt] \{x\}&\text{if}&x\in M\setminus N,\\ \end{array}\right.$$

$$\tilde{U}:=\mathrm{STR}(\tilde{N},M,\tilde{\rho })=\tilde{N}\cup M.$$

$$\alpha :=\mathrm{STR}(\lambda ,{\mathrm{id}}_M):\tilde{U}⟶{\tilde{U}}^{\prime }.$$

$${\tau }_{U,N}:\tilde{U}(N)⟶U.$$

$$\tilde{M}(N)\cong \mathrm{STR}(N,M,i)$$

$$\tilde{\alpha }:={\tilde{\alpha }}_{N^{\prime },N}:{\tilde{U}}^{\prime }(N^{\prime })⟶\tilde{U}(N),$$

$${\tilde{U}}^{\prime }(N^{\prime })\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\tilde{U}}^{\prime }(N^{\prime })=\mathrm{STR}(N^{\prime },M^{\prime },{\rho }^{\prime }),\tilde{U}(N)=\mathrm{STR}(N,M,\rho ).$$

$$\tilde{\alpha }:=\mathrm{STR}(\lambda ,\gamma ):{\tilde{U}}^{\prime }(N^{\prime })⟶\tilde{U}(N).$$

$$\tilde{\alpha }:U^{\prime }⟶\tilde{U}(N),$$

$$\widetilde{\alpha}(x^{\prime})=\left\{\begin{array}[]{lll}\widetilde{\alpha(x^{\prime})}&\text{if}&x^{\prime}\in\mathcal{T}(U^{\prime})_{0},\\[2.84526pt] \alpha(x^{\prime})&\text{if}&x^{\prime}\in eU^{\prime}.\\ \end{array}\right.$$

$$\tilde{\phi }:R⟶\tilde{U}(N),a\mapsto \widetilde{\phi (a)},$$

$$\alpha ({\mathcal{T}}^{\prime }(U{)}_0)=\alpha ({\phi }^{\prime }(R))=\phi (R)=N.$$

$$\tilde{\alpha }:U^{\prime }⟶\tilde{U}(N).$$

$$\tilde{\alpha }({\phi }^{\prime }(a))=[\alpha ({\phi }^{\prime }(a)){]}^{\sim }=\widetilde{\phi (a)}=\tilde{\phi }(a).$$

$${\alpha }_{\tilde{\phi },{\phi }^{\prime }}:U^{\prime }⟶\tilde{U}(N)$$

$$G(\psi ):=\psi (R)\cap M=\{\psi (a)|a\in R,\psi (a)=v(a)\}$$

$$xy=e\psi (a)\psi (b)=\psi (a)\psi (b)=\psi (ab).$$

$$\phi \le {\psi }_1\le \tilde{\phi },\phi \le {\psi }_2\le \tilde{\phi }.$$

$$\tilde{N}:=\mathcal{T}(\tilde{U}{)}_0=\tilde{\phi }(R),\tilde{N}\cap M=\{0\},$$

$$N:=\phi (R),N\cap M=G(\phi ),$$

Full text

1. 1 Lifting ghosts to tangibles
2. 2 The partial tangible lifts of an $m$-supervaluation
3. 3 Tangible and mixing transmissions
4. 4 The $(t,m)$-factorization of the composite of two transmissions
5. 5 Ideal-generating sections and tyrants
6. 6 Equalizers
7. 7 Producing isolated tangibles and tyrants

Introduction

This paper is a further step in the study of supertropical semirings, combining algebraic and categorical viewpoints, as a sequel of [IKR4]. As in [IKR1]–[IKR5], we aim for a better understanding of the commutative algebra over these semirings, especially with respect to monoid valuations (written m-valuations) and their refinement by supervaluations [IKR4,  §4]. These valuations generalize the classical valuations, whose targets, which are ordered abelian groups, are replaced by ordered monoids [IKR1, §2]. Classical valuations, with $\mathbb{R}$ as a target, are extensively used in tropical geometry and are at the heart of this theory. Approaching other important valuation targets, such as $\mathbb{N}$, has been our initial motivation for developing supervaluations, supported by a rich enough algebraic foundation [I1, I2], [IR1]–[IR5], [IKR5].

Any ordered monoid $M$ gives rise to a bipotent semiring $T(M)$ whose multiplication is the original monoid operation, and whose addition is the maximum in the given ordering. For example, the max-plus semiring $T(\mathbb{R})$, the underlying structure of tropical geometry, is obtained by taking the real numbers with the standard ordering and summation. Any bipotent semiring extends to a supertropical semiring $R$, in which $a+a=\nu(a)$ for any  $a\in R$, where $\nu:R\to\mathcal{G}(R)$, called the ghost map, is a projection on a distinguished deal $\mathcal{G}(R)$, called the ghost ideal of $R$. Equivalently, $a+a=ea$, where $e\in\mathcal{G}(R)$ is a distinguished idempotent element satisfying $eR=e\mathcal{G}(R)=\mathcal{G}(R)$. This addition, which replaces the rule $a+a=a$ in max-plus algebra, allows an algebraic capturing of combinatorial properties, where the ghost ideal takes the role of the zero element in classical algebra. The elements of $\mathcal{T}(R)=R\setminus\mathcal{G}(R)$ are termed tangible elements. With this setup, any bipotent semiring can be realized as a supertropical semiring having no tangible elements.

In supertropical setting tangible elements are the meaningful elements which essentially frame the central algebraic and geometric features within the theory. Factorization of tangibles is then a natural issue, rather delicate in this setup, especially since different $x_{1},\dots,x_{m}\in\mathcal{T}(R)$ may have the same ghost image $ex_{1}=\dots=ex_{m}$. Therefore, to characterize factorization of tangibles, the tangible preimage of $ex$ under the ghost map $\nu:R\to\mathcal{G}(R)$ needs a comprehensive study, including a categorical viewpoint, delivered by valuations and transmissions, as well as the generalization of supertropical semirings to supertropical monoids. Before delving into more explicit structural details, let us first give a motivating example.

Example**.**

Let $R$ be a supertropical semiring, where $eR=\{e,c,c^{2},\dots\}\ {\dot{\cup}}\ \{0\}$ is a totaly ordered monoid with different $c^{i}$, and $\mathcal{T}(R)$ is the free abelian monoid on two generators  $x,y$. Consider the ghost map $\nu:R\to eR$ that sends $x^{i}y^{j}$ to $c^{i+j}=ex^{i}y^{j}$, so that the tangible preimage of  $c^{n}$ consists of all $x^{i}y^{j}$ with $i+j=n$. Identifying two elements $z\sim z^{\prime}$ with $ez=ez^{\prime}$ by an equivalence relation $\sim$, which respects the semiring operations, any product  $zx$ is identified with $z^{\prime}x$ for every $x$. Consequently, factorizations of $zx$ coincide with $z^{\prime}x$ when quotienting by the given equivalence relation $\sim$. Therefore, since factorizations respect quotienting, they can be explored in terms of maps’ factorization which agrees with the ghost maps, where one needs a special care of particular pathologies arise in this setting.

With this approach to factorization, let us review the general context of supertropical structures and supervaluations, which links the supertropical theory to classical theory. Supervaluations refine classical valuations [IKR1]-[IKR3] by replacing their target semirings by supertropical semirings, and provide an enriched algebraic framework to tropical geometry. An m-valuation on a semiring $R$ is a multiplicative monoid homomorphism $v:R\to M$ to a bipotent semiring $M$ satisfying $v(x+y)\ {\leq}\ v(x)+v(y)$, cf. [IKR1, §2]. An m-valuation $v$ is a valuation, if $M\setminus\{0\}$ is a cancellative multiplicative monoid. For rings, these valuations coincide with the valuations defined by Bourbaki [B], and lead to a mapping of algebraic objects, called tropicalization. Examples of m-valuations on rings which are not valuations were given in [IKR1, §1].

To obtain a categorical framework which comprises supervaluations, in the category $\operatorname{STROP}$ of supertropical semirings (Definition 1.1), transmissions111Transmissions are called transmissive maps in [IKR1, §5]., which are more general than semiring homomorphisms, appear to be the “right” morphisms. A transmission $\alpha:R\to S$ is a multiplicative map whose restriction $eR\to eS$ to the ghost ideals is a semiring homomorphism [IKR4, Definition 1.4]. Transmissions are those maps whose composition $\alpha\circ v$ with any supervaluation $v$ is again a supervaluation [IKR1, §5]. Therefore, $\operatorname{STROP}$ includes bipotent semirings as a subcategory, and provides a richer algebraic setting for their study.

For a more comprehensive view, the category $\operatorname{STROP}$ is enlarged to the category $\operatorname{STROP}_{m}$ of supertropical monoids, which contains $\operatorname{STROP}$ as a full subcategory. A supertropical monoid $(U,\cdot\,)$ is a pointed monoid with a distinguished idempotent element $e=e_{U}$, for which the subset $eU$ carries a total ordering compatible with multiplication, such that $eU$ becomes a semiring by defining addition $+:eU\times eU\to eU$ as the maximum (Definition 1.2). As in the case of supertropical semirings, this addition extends to the entire  $U$ by the use of the multiplication  $\cdot$ and the idempotent element  $e$, but distributivity on $(U,\cdot\,,+)$ may fail. When it does not fail, $U$ is a supertropical semiring. A morphism $\alpha:U\to V$ in $\operatorname{STROP}_{m}$ is a transmission as defined for $\operatorname{STROP}$, which restricts to a homomorphism $eU\to eV$ of bipotent semirings, and obeys the rules TM1-TM5 of transmissions in [IKR1, Theorem 5.4].

Since a bipotent semiring $M$ can be viewed as a supertropical semiring having only ghost elements, where $e=1_{M},$ the category $\operatorname{STROP}_{m}/M$ of supertropical monoids over  $M$ may be viewed as the category of supertropical monoids $U$ having a fixed ghost ideal $eU=M$, whose morphisms are transmissions $\alpha:U\to V$ with $\alpha(x)=x$ for all $x\in M.$ The surjective transmissions over $M$ are called fiber contractions, as in the case of supertropical semirings [IKR1, §6]. If $\alpha:U\twoheadrightarrow V$ is a fiber contraction and $U$ is a supertropical semiring, then $V$ is also a supertropical semiring [IKR4, Theorem 1.6], and $\alpha$ is a semiring homomorphism [IKR1, Proposition 5.10.iii].

For every supertropical monoid $U$ there exists a fiber contraction $\sigma_{U}:U\to{\widehat{U}}$, where  ${\widehat{U}}$ is the supertropical semiring associated to $U$, such that every fiber contraction $\alpha:~{}U\twoheadrightarrow V$ factors (uniquely) through $\sigma_{U}$, i.e., $\alpha=\beta\circ\sigma_{U}$, where $\beta:{\widehat{U}}\to V$ is a fiber contraction. Namely, $\operatorname{STROP}/M$ is a full reflective subcategory of $\operatorname{STROP}_{m}/M$ [F,  p9], [FS, 1.813].

Universal problems appearing in $\operatorname{STROP}/M$ are generalized to $\operatorname{STROP}_{m}/M$ in an obvious way and can be solved, often more easily. Such solutions are delivered to $\operatorname{STROP}/M$ by reflections $\sigma_{U}:U\to{\widehat{U}}$. This approach pertains in particular to m-valuations $v:R\to M$ on a ring $R$, and to supervaluations $\varphi:R\to S$ with $S$ a supertropical semiring over $eS=M$, as defined in [IKR1, §4]. (They are also defined for $R$ a semiring.) Such a supervaluation applies to the coefficients of a (Laurent) polynomial $f(\lambda)\in R[\lambda]$ in a set $\lambda$ of $n$ variables, and gives a polynomial $f^{\varphi}(\lambda)$ over $S$. This view helps to analyze supertropical root sets and tangible components of polynomials $g(\lambda)\in F[\lambda]$, obtained from $f^{\varphi}(\lambda)\in S[\lambda]$ by passing from  $S$ to various supertropical semifields $F$ [IR1, §5 and  §7]. To this end, one needs a good control on the set $\{a\in R\ {|}\ \varphi(a)\in M\}\ {=}\ \{a\in R\ {|}\ \varphi(a)=v(a)\},$ which plays a central role in §1 and $\S\ref{sec:2}$ below.

Given an m-supervaluation $\varphi:R\to U$ covering an m-valuation $v:R\to M$, in §1 we construct a tangible m-supervaluation $\widetilde{\varphi}:R\to\widetilde{U}$ which is minimal such that $\widetilde{\varphi}\geq\varphi$ (Theorem 1.12). In §2 we then classify the m-supervaluations $\psi$ satisfying $\varphi\leq\psi\leq\widetilde{\varphi},$ called the partial tangible lifts of $\varphi.$ They are uniquely determined by their ghost value sets

[TABLE]

cf. Theorem 2.4. These sets are ideals of the semiring $M,$ and all ideals $\mathfrak{a}\subset G(\varphi)$ occur in this way (Theorem 2.7). Unfortunately, the ghost value set $G(\psi)$ does not control the set $\{a\in R\ {|}\ \psi(a)\in M\}$ completely. We only able to state that this set is contained in the preimage $v^{-1}(G(\psi))$. If $\varphi$ is a supervaluation, then $\widecheck{\varphi}:=(\widetilde{\varphi})^{\wedge}$ is the supervaluation, which is a partial tangible lift of $\varphi$ having smallest ghost value set.

In §3 we undertake a fine analysis of surjective transmissions $\alpha:U^{\prime}\twoheadrightarrow U$ for supertropical monoids $U^{\prime},U$. Such transmissions can be approached via TE-relations, i.e., equivalence relations on $U^{\prime}$, compatible with its monoid structure in the appropriate sense [IKR4,  Definition  1.7]. Namely, given a transmission $\alpha:U^{\prime}\twoheadrightarrow U$, it induces the equivalence relation $E=E(\alpha)$ on $U^{\prime}$ which identifies all elements $x^{\prime}\in U^{\prime}$ having the same image under $\alpha$. Then $\alpha$ factorizes as $\alpha=\rho\circ\pi_{E}$, where  \rho:U^{\prime}/E\xrightarrow{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}U is an isomorphism and $\pi_{E}:U^{\prime}\twoheadrightarrow U^{\prime}/E$ is the canonical surjection sending each $x^{\prime}\in U^{\prime}$ to its equivalent class  $[x^{\prime}]_{E}$.

Dividing out the zero kernel of a transmission $\alpha:U^{\prime}\to U$ [IKR4, §1], without loss of generality we assume that $\alpha^{-1}(0)=\{0^{\prime}\}$. A transmission $\alpha$ is called tangible (Definition  1.2), if $\alpha(\mathcal{T}(U^{\prime}))\subset\mathcal{T}(U)\cup\{0\}$. It is called mixing, if for any different elements $x^{\prime}_{1},x^{\prime}_{2}\in U^{\prime}$ with $\alpha(x^{\prime}_{1})=\alpha(x_{2}^{\prime})$ there exists a mixing element $z^{\prime}\in U^{\prime}$, might interchanging $x_{1}^{\prime}$ and $x_{2}^{\prime}$, such that $x_{1}^{\prime}z\in\mathcal{T}(U^{\prime})$ and $x_{2}^{\prime}z\in eU^{\prime}$. We prove that every surjective transmission $\alpha:U^{\prime}\twoheadrightarrow U$ with trivial kernel factorizes as

[TABLE]

where $\alpha_{t}:U^{\prime}\twoheadrightarrow V$ is a tangible transmission, and $\alpha_{m}:V\twoheadrightarrow U$ is a mixing transmission which covers the identity of $eU$, and thus is a multiplicative fiber contraction of $V$ over $eV=eU$.

Choosing as small as possible $V$ in the appropriate sense, the factorization (0.1) is unique (Theorem 3.10.(ii)), called the $(t,m)$-factorization of $\alpha$. The factor $\alpha_{m}$ can be replaced by the surjection $\pi_{E}:V\twoheadrightarrow V/E$ of a suitable MFCE-relation (=multiplicative fiber conserving equivalence) $E$ on $eV=eU$. This indicates that finding $(t,m)$-factorizations is essentially a matter of MFCE-relations, and so can be approached explicitly by the methods developed in [IKR1]–[IKR4]. In §4 we determine the $(t,m)$-factorizations of the composite $\beta\circ\alpha$ of two surjective transmissions $\alpha$ and $\beta$ having trivial zero kernel.

The idea beyond $\S\ref{sec:3}$ and $\S\ref{sec:4}$ is to study the “fate” of a given (tangible) family $\{x_{1},\dots,x_{r}\}$ in a supertropical monoid $U$, satisfying $ex_{1}=\cdots=ex_{r}$, due to multiplying by some element $u\in U$. This means to decide whether $x_{i}u\in\mathcal{T}(U)$ or $x_{i}u\in eU$, i.e., the products becomes a ghost or zero. The latter case with $u\in\mathcal{T}(U)$ is of more interest. We follow this idea in  §5–§7, relying on hierarchy of elements which is phrased by a patriarchal terminology.

An element $z\in\mathcal{T}(U)$ is called a son of $x\in\mathcal{T}(U)$ over $c=ex$, if $ez=c$ and there exists $u\in\mathcal{T}(U)$ such that $z=xu$. Taking a patriarchal viewpoint, $z$ is regarded as the outcome  $xu$ of pairing $x$ to some $u\in\mathcal{T}(U)$ such that $exu=c$, without specifying explicitly the “mother” element $u$ of $z$. In other words, $x$ and $z$ have the same ghost image $c$, where $z=xu$ for some unspecified tangible element $u$. An element $x$ is called a tyrant (or said to be tyrannic), if it has at most one son over $c\in eU$. It is said to be isolated, if it has no son  $z\neq x$ with $ex=ez$. The latter property is weaker than tyrannic, but to our opinion deserves similar attention.

For any tyrant $x$ in a supertropical monoid $U$ the set $\mathfrak{a}=eU\cup xU$ is obviously an ideal of $U$ containing $M=eU$. This ideal is small in the sense, that each fiber of the restricted ghost map $\nu_{U}|\mathfrak{a}$ has at most two elements, $\mathfrak{a}_{c}=\{c,z\}$ if $x$ has a son $z$ over $c$, otherwise $\mathfrak{a}_{c}=\{c\}$. Conversely, given an ideal $\mathfrak{a}\supset M$ with $|\mathfrak{a}_{c}|\leq 2$ for every $c\in M$, then $\nu_{U}:U\to M$ has a section $s:M\to U$ for which $\mathfrak{a}=M\cup s(M)$. As explicated in §5, it follows that the MFCE-relations, for which every class contains at most one tangible element, are the compressions of these ideals $M\cup s(M)$, cf. [IKR4,  Definition  2.5].

The equalizer $\operatorname{Eq}(S)$ of a subset $S$ of $U_{a}=\{x\in U\ {|}\ ex=a\}$ is defined as the finest equivalence relation $E$ on $U$ with $s\sim_{E}t$ for all $s,t\in S$. More generally, given an arbitrary subset $S$ of $U$, the fiberwise equalizer $\operatorname{Feq}(S)$ of $U$ is the finest equivalence relation $E$ on $U$ for which $s\sim_{E}t$ for any $s,t\in S$ with $es=et$. $\operatorname{Eq}(S)$ and $\operatorname{Feq}(S)$ are obtained in §6 by explicit construction, using the so-called “$S$-pathes” in $U$. These pathes serve as our main technical tool in §7.

The universal MFCE-relation $\operatorname{T}(x)$ turning a given $x\in\mathcal{T}(U)$ into a tyrant is thoroughly studied in  §7. This means that the relation $\operatorname{T}(x)$ is the finest MFCE-relation $E$ on $U$ for which  $[x]_{E}$ is tyrannic on  $U/E$, provided that such relation $E$ exists on $U$, and otherwise $[x]_{\operatorname{T}(x)}$ is ghost in $U/\operatorname{T}(x)$. This relation $\operatorname{T}(x)$ is the fiberwise equalizer $\operatorname{Feq}(\SS(x))$ of the set $\SS(x)=(Ux)\cap\mathcal{T}(U)$ of sons of $x$. Using the results of §6, we gain an explicit criterion when $[x]_{\operatorname{T}(x)}$ is a tyrant and when is not (Theorem 7.6).

In an analogous way we obtain the universal MFCE-relation $\operatorname{Is}(x)$ which isolates $x$ in its fiber $U_{ex}$, as well as the universal MFCE-relation $\operatorname{Sis}(x)$ which isolates every son  $z$ of $x$ in its fiber  $U_{ez}$, together with criteria for the existence of such MFCE-relations (Theorems  7.9 and  7.16). It turns out, perhaps astonishing at first glance, that under a mild cancellation hypothesis on $eU$ these MFCE-relations coincide (Theorem 7.18) and $\operatorname{Is}(x)=\operatorname{Sis}(x)$.

The results in §3–§7 may be viewed as instances of a supertropical divisibility theory, performed upon TE-relations instead of ideals, in particular prime ideals, in classical commutative algebra. In the same spirit, prime and radical ideals can be replaced by special equivalence relations to develop a systematic theory of commutative $\nu$-algebra  [I2], which generalizes supertropical algebra.

1. Lifting ghosts to tangibles

We recall our underlying structures.

Definition 1.1**.**

A supertropical semiring is a semiring $R$ where $e:=1+1$ is idempotent (i.e., $e+e=e$) such that, for all  $a,b\in R$, $a+b\in\{a,b\}$ whenever $ea\neq eb$ and $a+b=ea$ otherwise. This implies $ea=0\ {\Rightarrow}\ a=0.$ The target of the ghost map $\nu:a\mapsto ea$ is the ghost ideal $eR$ of $R$, which is a bipotent semiring (with unit element   $e$), i.e., $a+b$ is either $a$ or $b$, for any $a,b\in eR$. The semiring $eR$ is totally ordered by the rule

[TABLE]

$\mathcal{T}(R):=R\setminus(eR)$* is the set of tangible elements of $R$, and $\mathcal{G}(R):=eR\setminus\{0\}$ is the set of nonzero ghost elements. The zero $0=e0$ is regarded mainly as a ghost. *

More generally, we have the following structure.

Definition 1.2**.**

A supertropical monoid $U$ is an abelian monoid $(U,\cdot\,)$ with an absorbing element $0:=0_{U}$, i.e., $0\cdot x=0$ for every $x\in U$, and a distinguished idempotent $e:=e_{U}$ such that

[TABLE]

In addition, the submonoid $M:=eU$ of $U$ is equipped with a total ordering, compatible with multiplication [IKR4, Definition 1.1], which is again determined by rule (1.1). The map $\nu_{U}:U\to M$, $x\mapsto ex$, is a monoid homomorphism, called the ghost map of $U$. Tangible elements $\mathcal{T}:=\mathcal{T}(U)$ and ghost elements $\mathcal{G}:=\mathcal{G}(U)$ are defined exactly as in Definition  1.1.

A supertropical monoid $U$ is called unfolded, if the set $\mathcal{T}(U)_{0}:=\mathcal{T}(U)\cup~{}\{0\}$ is closed under multiplication.

If $U$ is unfolded, then $N:=\mathcal{T}(U)_{0}$ is a multiplicative monoid with absorbing element [math]. Furthermore, $M:=eU$ is a totally ordered monoid with absorbing element [math], and the restriction

[TABLE]

is a monoid homomorphism with $\rho^{-1}(0)=\{0\}$. Observing that $e_{U}=1_{M}=\rho(1_{N})$, we see that the supertropical monoid $U$ is completely determined by the triple $(N,M,\rho)$. This paves a way to constructing all unfolded supertropical monoids up to isomorphism.

Construction 1.3**.**

Given a totally ordered monoid $M$ with absorbing element $0_{M}\leq x$ for all $x\in M$, i.e., a bipotent semiring $M$, let $N$ be an (always commutative) monoid with absorbing element $0_{N}$, together with a multiplicative map $\rho:N\to M$ with $\rho(1_{N})=1_{M}$, $\rho^{-1}(0_{M})=\{0_{N}\}$. We define an unfolded supertropical monoid $U$ to be the disjoint union of $M\setminus\{0_{M}\}$, $N\setminus\{0_{N}\}$ and a new element [math], identified as $0=0_{M}=0_{N}$. We write

[TABLE]

The multiplication on $U$ is given by the rules, in obvious notation,

[TABLE]

It is easy to verify that $(U,\cdot\,)$ is a (commutative) monoid with $1_{U}=1_{N}$ and absorbing element [math]. Let $e:=1_{M}$, then $eU=M$ and $\rho(x)=ex$ for $x\in M$. Furthermore, $ex=0$ iff $x=0$ for any $x\in U$, since $\rho^{-1}(0)=\{0\}$. Thus $(U,\cdot\,,e)$, together with the given ordering on $M=eU$, is a supertropical monoid, clearly unfolded. We denote the supertropical monoid $U$ by $\operatorname{STR}(N,M,\rho)$.222The notation $\operatorname{STR}(N,M,\rho)$ differs slightly from the notation $\operatorname{STR}(\mathcal{T},\mathcal{G},v)$ in [IKR1, Construction  3.16], causes no confusion regarding the ambient context.

This construction generalizes the construction of supertropical domains [IKR1] (loc. cit. Construction 3.16). There, to obtain all supertropical predomains up to isomorphism, it was assumed that $N\setminus\{0\}$ and $M\setminus\{0\}$ are closed under multiplication, and that the monoid $M\setminus\{0\}$ is cancellative. Omitting only the cancellation hypothesis would give us a class of supertropical monoids not broad enough for our work below.

We add a description of the transmissions between two unfolded supertropical monoids.

Proposition 1.4**.**

Assume that $U^{\prime}=\operatorname{STR}(N^{\prime},M^{\prime},\rho^{\prime})$ and $U=\operatorname{STR}(N,M,\rho)$ are unfolded supertropical monoids.

1. (i)

If $\lambda:N^{\prime}\to N$ is a monoid homomorphism with $\lambda(0)=0$, $\mu:M^{\prime}\to M$ is a semiring homomorphism, and $\rho^{\prime}\lambda=\mu\rho$, then the well-defined map

[TABLE]

sending $x^{\prime}\in N^{\prime}$ to $\lambda(x^{\prime})$ and $y^{\prime}\in M^{\prime}$ to $\mu(y^{\prime})$, is a tangible transmission **[IKR4,  Definition 2.3]**. 2. (ii)

In this way we obtain all tangible transmissions from $U^{\prime}$ to $U$.

Proof.

(i): A straightforward check. (ii): Obvious. ∎

Given an m-valuation $v:R\to M$ with support $v^{-1}(0)=\mathfrak{q}$, the supertropical semiring  $U^{0}(v)$ appearing in [IKR4, Theorem 7.4] may be viewed as an instance of Construction  1.3, as follows.

Example 1.5**.**

Let $E$ denote the equivalence relation on $R$ having the equivalence classes $[0]_{E}=\mathfrak{q}$ and $[x]_{E}=\{x\}$ for $x\in R\setminus\mathfrak{q}$. $E$ is multiplicative, and hence $R/E$ is a monoid with absorbing element $[0]_{E}=0$, identified with the subset $(R\setminus\mathfrak{q})\cup\{0\}$ in the obvious way. The map $v:R\to M$ induces a monoid homomorphism ${\bar{v}}:R/E\to M$ given by ${\bar{v}}(x)=v(x)$ for $x\in R\setminus\mathfrak{q}$, ${\bar{v}}(0)=0$. Thus ${\bar{v}}^{-1}(0)=\{0\}$ and

[TABLE]

We next aim for “unfolding” methods of an arbitrary supertropical monoid $U$. By this we roughly mean a fiber contraction $\tau:\widetilde{U}\to U$, where $\widetilde{U}$ is an unfolded supertropical monoid and the fibers $\tau^{-1}(x)$, $x\in U$, are as small as possible. More precisely we decree

Definition 1.6**.**

Let $M:=eU$, and let $N$ be a submonoid of $(U,\cdot\;)$ which contains the set $\mathcal{T}(U)_{0}$. An unfolding of $U$ along $N$ is a fiber contraction $\tau:\widetilde{U}\to U$ over $M$ (in particular $e\widetilde{U}=M$), such that

[TABLE]

where $\tilde{x}\in\mathcal{T}(U)_{0}$. For any $x\in N$, we call $\tilde{x}$ the tangible lift of $x$ (with respect to $N$).

Notice that this forces $\tau(\mathcal{T}(\widetilde{U})_{0})=N$, and moreover for any $x\in N$ the tangible fiber $\tilde{x}$ is the unique element of $\mathcal{T}(\widetilde{U})_{0}$ with $\tau(\tilde{x})=x$, hence $\mathcal{T}(\widetilde{U})_{0}=\{\tilde{x}\ {|}\ x\in N\}$. Thus, if $\tau:\widetilde{U}\to U$ is an unfolding along $N$, then the restriction $\mathcal{T}(\widetilde{U})_{0}\to N$, $\tilde{x}\mapsto x$, of $\tau$ is a monoid isomorphism, and $\tau$ itself is an ideal compression with ghost kernel ${(M\cap N)}^{\sim}\cup M$ (cf. [IKR4, Definition 1.5]), where ${(M\cap N)}^{\sim}:=\{\tilde{x}\ {|}\ x\in M\cap N\}$.

Theorem 1.7**.**

**

1. (i)

Given a pair $(U,N)$ consisting of a supertropical monoid $U$ and a multiplicative submonoid $N\supset\mathcal{T}(U)_{0}$, there exists an unfolding $\tau:\widetilde{U}\to U$ of $U$ along $N$. 2. (ii)

If $\tau^{\prime}:\widetilde{U}^{\prime}\to U$ is a second unfolding of $U$ along $N$, then there exists a unique isomorphism of supertropical monoids $\alpha:\widetilde{U}\ {\tilde{\to}}\ \widetilde{U}^{\prime}$ with $\tau^{\prime}\circ\alpha=\tau$.

Proof.

(i) Existence: Since $M$ is an ideal of $U$, the set $M\cap N$ is a monoid ideal of $N$. We have $U=N\cup M$, since $N\supset\mathcal{T}(U)_{0}$. Let $\rho:N\to M$ denote the restriction of $\nu_{U}$ to  $N$. It is a monoid homomorphism with $\rho^{-1}(0)=\{0\}$.

Let $\widetilde{N}$ denote a copy of the monoid $N$ with copying isomorphism $x\mapsto\tilde{x}$ ($x\in N$), and let $\widetilde{\rho}:\widetilde{N}\to M$ denote the monoid homomorphism from $\widetilde{N}$ to $M$ corresponding to $\rho:N\to M$. Thus $\widetilde{\rho}(\tilde{x})=\rho(x)=ex$ for $x\in N$. Define the unfolded supertropical monoid

[TABLE]

In $\widetilde{U}$ we have $\widetilde{0}_{U}=0$ and $\widetilde{N}\cap M=\{0\}$. Further $\mathcal{T}(\widetilde{U})_{0}=\widetilde{N}$ and $e\widetilde{U}=eU=M$.

We obtain a well-defined surjective map $\tau:\widetilde{U}\to U$ by setting $\tau(\tilde{x}):=x$ for $x\in N$, $\tau(y):=y$ for $y\in M$. As easily checked, $\tau$ is multiplicative, sending [math] to [math], $1\in\mathcal{T}(\widetilde{U})$ to $1\in N$, which restricts to the identity on $M$. Thus $\tau$ is a fiber contraction [IKR4, Definition 2.1]. The fibers of $\tau$ are as indicated in Definition 1.6; hence $\tau$ is an unfolding of $U$ along $N$. (ii) Uniqueness: Let $\widetilde{\tau}:\widetilde{U}\to U$ and $\widetilde{\tau}^{\prime}:\widetilde{U}^{\prime}\to U$ be unfoldings of $U$ along $N$ with tangible lifts $x\mapsto\tilde{x}$ and $x\mapsto\tilde{x}^{\prime}$ respectively. Without loss of generality we assume that $\widetilde{U}=\operatorname{STR}(\widetilde{N},M,\widetilde{\rho})$ and $\widetilde{U}^{\prime}=\operatorname{STR}(\widetilde{N}^{\prime},M,\widetilde{\rho}^{\prime})$ with tangible lifts $x\mapsto\tilde{x}$ and $x\mapsto\tilde{x}^{\prime}$ ($x\in N$). Then $\widetilde{\rho}(\tilde{x})=\widetilde{\rho}^{\prime}(\tilde{x}^{\prime})=ex$ for every $x\in N$. The map $\lambda:\widetilde{N}\to\widetilde{N}^{\prime}$, given by $\lambda(\tilde{x})=\tilde{x}^{\prime}$ for $x\in N$, is a monoid isomorphism with $\widetilde{\rho}^{\prime}\circ\lambda=\operatorname{id}_{M}\circ\widetilde{\rho}$. Thus we have a well defined transmission (cf. Proposition 1.4)

[TABLE]

$\alpha$ is an isomorphism over $U$, i.e., an isomorphism with $\tau^{\prime}\circ\alpha=\tau$, clearly the only one. ∎

Notation 1.8**.**

We call the map $\tau:\widetilde{U}\to U$ constructed in part (i) of the proof of Theorem 1.7 “the” unfolding of $U$ along $N$ and, when necessary, write it more precisely as

[TABLE]

Sometimes we abusively denote any unfolding of $U$ along $N$ in this way, justified by Theorem 1.7.(ii).

Example 1.9**.**

In the special case that $U=eU=M$, $N$ can be any submonoid of $M$ containing [math]. Then $\widetilde{M}(N)=\widetilde{N}\cup M$ with $\widetilde{N}\cap M=\{0\}$, and

[TABLE]

with $i:N\hookrightarrow M$ the inclusion mapping. For every $x\in N$ there exists a unique tangible element $\tilde{x}$ in $\widetilde{M}(N)$ with $e\tilde{x}=x$, while for $x\in M\setminus N$ there exists no such element.

Theorem 1.10**.**

Assume that $\alpha:U^{\prime}\to U$ is a transmission of supertropical monoids [IKR4, Definition 1.4], and that $N^{\prime}\supset\mathcal{T}(U^{\prime})_{0}$, $N\supset\mathcal{T}(U)_{0}$ are submonoids of $U^{\prime}$ and $U$ with $\alpha(N^{\prime})\subset N$. Then there exists a unique tangible transmission

[TABLE]

called the tangible unfolding of $\alpha$ along $N^{\prime}$ and $N$, such that the diagram

[TABLE]

commutes.

Proof.

Let $M^{\prime}:=eU^{\prime}$, $M:=eU$, and let $\rho^{\prime}:N^{\prime}\to M$, $\rho:N\to M$ denote the monoid homomorphisms obtained from $\nu_{U^{\prime}}$ and $\nu_{U}$ by restriction to $N^{\prime}$ and $N$. Then

[TABLE]

The map $\alpha$ restricts to monoid homomorphisms $\lambda:N^{\prime}\ {\to}\ N$ and $\gamma:M^{\prime}\to M$ with $\lambda(0)=0$, $\gamma(0)=0$, where $\gamma$ is order preserving. So $\gamma\circ\nu_{U^{\prime}}=\nu_{U}\circ\alpha$, hence $\gamma\circ\rho^{\prime}=\rho\circ\lambda$, and we obtain the tangible transmission

[TABLE]

Clearly $\tau_{U,N}\circ\widetilde{\alpha}=\alpha\circ\tau_{U^{\prime},N^{\prime}}$. Since any tangible transmission from $\widetilde{U}^{\prime}(N^{\prime})$ to $\widetilde{U}(N)$ maps  $\widetilde{N}^{\prime}$ to $\widetilde{N}$ and $M^{\prime}$ to $M$, it is evident that $\widetilde{\alpha}$ is the only such map. ∎

Corollary 1.11**.**

Assume that $\alpha:U^{\prime}\to U$ is a tangibly surjective transmission of supertropical monoids, i.e., $\mathcal{T}(U)\subset\alpha(\mathcal{T}(U^{\prime}))$, and $U^{\prime}$ is unfolded. Let $N:=\alpha(\mathcal{T}(U^{\prime})_{0})$, which is a submonoid of $U$ containing $\mathcal{T}(U)_{0}$.

1. (i)

There exists a unique tangible transmission

[TABLE]

called the tangible lift of $\alpha$, such that $\tau_{U,N}\circ\widetilde{\alpha}=\alpha.$ 2. (ii)

If $x^{\prime}\in U^{\prime}$, then

[TABLE]

Proof.

(i): Apply Theorem 1.10 with $N^{\prime}:=\mathcal{T}(U^{\prime})_{0}$, and observe that $\widetilde{U}^{\prime}(N^{\prime})=U^{\prime}$, since  $U^{\prime}$ is unfolded. (ii): Obvious, since $\tau_{U,N}(\widetilde{\alpha}(x^{\prime}))=\alpha(x^{\prime})$ and $\widetilde{\alpha}(x^{\prime})\in\mathcal{T}(\widetilde{U})_{0}$ iff $x^{\prime}\in\mathcal{T}(U^{\prime})_{0}$. ∎

We are ready to construct “tangible lifts” of m-supervaluations.

Theorem 1.12**.**

Assume that $\varphi:R\to U$ is a tangibly surjective m-supervaluation, i.e., $\mathcal{T}(U)\subset\varphi(R)$ {e.g. $\varphi$ is surjective; $U=\varphi(R)\cup e\varphi(R)$}. Let $N:=\varphi(R)$. It is a submonoid of $U$ containing $\mathcal{T}(U)$.

1. (i)

The map

[TABLE]

with $\widetilde{\varphi(a)}$ denoting the tangible lift of $\varphi(a)$ w.r.t. $N$, is a tangible m-supervaluation of $\varphi$, called the tangible lift of $\varphi$ **[IKR4, Definition 2.3]**. 2. (ii)

If $\varphi^{\prime}:R\to U^{\prime}$ is a tangible m-supervaluation dominating $\varphi$, then $\varphi^{\prime}$ dominates $\widetilde{\varphi}$.

Proof.

(i): $\widetilde{\varphi}$ is multiplicative, $\widetilde{\varphi}(0)=0$, $\widetilde{\varphi}(1)=1$, and $e\widetilde{\varphi}=e\varphi$ is an m-valuation. Thus  $\widetilde{\varphi}$ is an m-supervaluation. By construction, $\widetilde{\varphi}$ is tangible. (ii): We may assume that the m-supervaluation $\varphi^{\prime}:R\to U^{\prime}$ is surjective, and hence $\varphi^{\prime}(R)\supset\mathcal{T}^{\prime}(U)_{0}$. Since $\varphi^{\prime}$ is tangible, this forces $\varphi^{\prime}(R)=\mathcal{T}^{\prime}(U)_{0}$. Thus $\mathcal{T}^{\prime}(U)_{0}$ is a submonoid of $U^{\prime}$, i.e., $U^{\prime}$ is unfolded. Since $\varphi^{\prime}$ dominates $\varphi$, there exists a transmission $\alpha:U^{\prime}\to U$ with $\varphi=\alpha\circ\varphi^{\prime}$, such that

[TABLE]

Thus we have the tangible lift of $\alpha$,

[TABLE]

So, for any $a\in R,$

[TABLE]

Thus $\widetilde{\varphi}=\widetilde{\alpha}\circ\varphi^{\prime},$ which proves that $\varphi^{\prime}$ dominates $\widetilde{\varphi}.$ ∎

Addendum 1.13**.**

As the proof has shown, if the m-valuation $\varphi^{\prime}$ is surjective, then $U^{\prime}$ is unfolded, and the transmission [IKR1, Definition 5.3]

[TABLE]

is the tangible lift of $\alpha_{\varphi,\varphi^{\prime}}:U^{\prime}\to U$.

Corollary 1.14**.**

If $\varphi$, $\psi$ are m-supervaluations covering $v$ and $\varphi\leq\psi$, then $\widetilde{\varphi}\leq\widetilde{\psi}$.

Proof.

We have $\varphi\leq\psi\leq\widetilde{\psi}$, and from Theorem 1.12.(ii) it follows that $\widetilde{\varphi}\leq\widetilde{\psi}$. ∎

2. The partial tangible lifts of an

$m$-supervaluation

In what follows $v:R\to M$ is a fixed m-valuation and $\varphi:R\to U$ is a tangible surjective m-supervaluation covering $v$. (Often $v$ and $\varphi$ will both be surjective.) The tangible lift $\widetilde{\varphi}:R\to\widetilde{U}$ (cf. Theorem 1.12) was introduced in §1, and now we strive for an explicit description of the m-supervaluations $\psi$ covering $v$ with $\varphi\leq\psi\leq\widetilde{\varphi}$.

Definition 2.1**.**

Given an m-supervaluation $\psi:R\to V$ covering $v:R\to M$, we call

[TABLE]

the ghost value set of $\psi$. {Notice that $eV=M$.}

Lemma 2.2**.**

Let $\psi_{1},\psi_{2}$ be m-supervaluations covering $v$. If $\psi_{1}\geq\psi_{2}$, then $G(\psi_{1})\subset~{}G(\psi_{2})$. If $\psi_{1}\sim\psi_{2}$, then $G(\psi_{1})=G(\psi_{2})$.

Proof.

Let $a\in R$. If $\psi_{1}\geq\psi_{2}$, then $\psi_{1}(a)\in M$ implies that $\psi_{2}(a)\in M$, due to dominance, condition ${\operatorname{D3}}$ in [IKR4, Definition 7.2]. Thus, $\psi_{1}(a)\in M$ iff $\psi_{2}(a)\in M$, for $\psi_{1}\sim\psi_{2}$. ∎

Lemma 2.3**.**

Assume that the m-valuation $v:R\to M$ is surjective. Then the ghost value set $G(\psi)$ of any m-supervaluation $\psi$ covering $v$ is an ideal of the semiring $M$.

Proof.

For $x\in G(\psi)$, $y\in M$ there exist $a,b\in R$ with $\psi(a)=x$, $e\psi(b)=y$, implying that

[TABLE]

Thus $xy\in\psi(R)\cap M=G(\psi)$, which proves that $G(\psi)\cdot M\subset G(\psi)$. Since $M$ is bipotent, $G(\psi)$ is also closed under addition. ∎

Theorem 2.4**.**

Assume that $\varphi:R\to U$ is an m-supervaluation covering $v:R\to M$, and that $\psi_{1},\psi_{2}$ are m-supervaluations covering $v$ with

[TABLE]

1. (i)

$\psi_{1}\geq\psi_{2}\quad{\Leftrightarrow}\quad G(\psi_{1})\subset G(\psi_{2})$; 2. (ii)

$\psi_{1}\sim\psi_{2}\quad{\Leftrightarrow}\quad G(\psi_{1})=G(\psi_{2})$.

Proof.

Without loss of generality, assume that $\varphi$ is surjective. Then also the m-super-valuations $\psi_{1},$ $\psi_{2},$ $\widetilde{\varphi}$ are surjective and, by Corollary 1.14, the tangible lifts $\widetilde{\psi}_{1}$ and $\widetilde{\psi}_{2}$ are both equivalent to $\widetilde{\varphi}$.

Again, without loss of generality, assume that $\varphi=\widetilde{\varphi}/E:=\pi_{E}\circ\widetilde{\varphi}$ with $E$ an MFCE-relation on $\widetilde{U}$, and that $\psi_{i}=\widetilde{\varphi}/E_{i}$ with an MFCE-relation $E_{i}$ ($i=1,2$). Let us describe these relations $E,E_{1},E_{2}$ explicitly. We have $\widetilde{U}=\widetilde{N}\cup M$, with

[TABLE]

furthermore $U=N\cup M$ with

[TABLE]

and a copying isomorphism

[TABLE]

of monoids (new notation!), which sends each $x\in N$ to its tangible lift $\tilde{x}$, as explained in  §1 (Definition 1.6, Proof of Theorem 1.7.i). Notice that $es(x)=x$ for $x\in N\cap M=G(\varphi).$

The relation $E$ has the 2-point equivalence classes $\{x,s(x)\}$ with $x$ running through  $G(\varphi)$, while all other $E$-equivalence classes are one-point sets. Analogously, $E_{i}$ has the 2-point set equivalence classes $\{x,s(x)\}$ with $x$ running through $G(\psi_{i})\subset G(\varphi)$, while again all other $E$-equivalence classes are one-point sets. Thus it is obvious that333As in [IKR1] we view an equivalence relation on a set $X$ as a subset of $X\times X$ in the usual way. $E_{1}\subset E_{2}$ iff $G(\psi_{1})\subset G(\psi_{2})$. But $E_{1}\subset E_{2}$ means that $\psi_{1}\geq\psi_{2}$. This gives claim (i), and claim (ii) follows. ∎

Definition 2.5**.**

We call the monoid isomorphism

[TABLE]

i.e., the copying isomorphism $s:N\tilde{\to}\widetilde{N}$ from the proof of Theorem 2.4, the tangible lifting map of $\varphi$.

Note that $s(x)y=s(xy)$ for $x\in\varphi(R)$, $y\in\mathcal{T}(\widetilde{U})_{0}$.

Henceforth we assume that the m-valuation $v:R\to M$ is surjective and that $\varphi:R\to U$ is a surjective m-supervaluation with $e\varphi=v$. The question arises whether every ideal $\mathfrak{a}$ of $M$ with $\mathfrak{a}\subset G(\varphi)$ appears as the ghost value set $G(\psi)$ of some m-supervaluation $\psi$ covering $v$ with $\varphi\leq\psi\leq\widetilde{\varphi}$. This is indeed true.

Construction 2.6**.**

We employ the tangible lifting map $s:\varphi(R)\ {\to}\ \widetilde{\varphi}(R)=\mathcal{T}(\widetilde{U})_{0}$ defined above. Assume that $\mathfrak{a}$ is an ideal of $M$ contained in $G(\varphi)$. We have

[TABLE]

since $s(x)y=s(xy)\in s(\mathfrak{a})$ for $x\in\mathfrak{a}$ and $y\in\mathcal{T}(U)$. We conclude that $s(\mathfrak{a})\cup M$ is an ideal of $U$. Let

[TABLE]

and $\widetilde{U}_{\mathfrak{a}}:=\widetilde{U}/E_{\mathfrak{a}}$. We regard $\widetilde{U}_{\mathfrak{a}}$ as a subset of $\widetilde{U}$, as indicated in [IKR4, Convention 3.3.a]. The map $\pi_{E_{\mathfrak{a}}}:\widetilde{U}\twoheadrightarrow\widetilde{U}_{\mathfrak{a}}$ is the ideal compression with ghost kernel $s(\mathfrak{a})\cup M$, and

[TABLE]

is an m-supervaluation. For any $a\in R$

[TABLE]

Clearly $\varphi\leq\widetilde{\varphi}_{\mathfrak{a}}\leq\widetilde{\varphi}$ and $G(\widetilde{\varphi}_{\mathfrak{a}})=\mathfrak{a}$. We call $\widetilde{\varphi}_{\mathfrak{a}}$ the tangible lift of $\varphi$ outside $\mathfrak{a}$, and we call any such map $\widetilde{\varphi}_{\mathfrak{a}}$ a partial tangible lift of $\varphi$.

Let $[\varphi,\widetilde{\varphi}]$ denote the “interval” of the poset $\operatorname{Cov}_{m}(v)$ containing all classes $[\psi]$ with $\varphi\leq\psi\leq\widetilde{\varphi}$, and let $[0,G(\varphi)]$ be the set of ideals $\mathfrak{a}$ of $M$ with $\mathfrak{a}~{}\subset~{}G(\varphi)$, ordered by inclusion. By Lemmas 2.2 and 2.3 we have a well defined order preserving map

[TABLE]

sending each class $[\psi]\in[\varphi,\widetilde{\varphi}]$ to the ideal $G(\psi)$. By Theorem 2.4 this map is injective, and by Construction 2.6 we know that it is also surjective. Thus we we have proved

Theorem 2.7**.**

The map

[TABLE]

is a well defined order preserving bijection. The inverse of this map sends an ideal $\mathfrak{a}\subset G(\varphi)$ to the class $[\widetilde{\varphi}_{\mathfrak{a}}]$ of the tangible lift of $\varphi$ outside $\mathfrak{a}$.

The poset $\operatorname{Cov}_{m}(v)$ is a complete lattice (cf. [IKR4, Corollary 7.5]). The poset $I(M)$ consisting of the ideals of $M$ and ordered by inclusion, is a complete lattice as well. Indeed, the infimum of a family $(\mathfrak{a}_{i}\ {|}\ i\in I)$ in $I(M)$ is the ideal $\bigcap_{i}\mathfrak{a}_{i}$, while the supremum is the ideal $\sum_{i}\mathfrak{a}_{i}=\bigcup_{i}\mathfrak{a}_{i}.$ {Recall once more that every subset of $M$ is closed under addition.} The intervals $[\varphi,\widetilde{\varphi}]$ and $[0,G(\varphi)]$ are again complete lattices, and thus the map $[\varphi,\widetilde{\varphi}]\ {\to}\ [0,G(\varphi)]$ in Theorem 2.7 is an anti-isomorphism of complete lattices. This implies the following

Corollary 2.8**.**

Assume that $(\psi_{i}\ {|}\ i\in I)$ is a family of supervaluations covering $v$ with $\varphi\leq\psi_{i}\leq\widetilde{\varphi}$ for each $i\in I$. Let $\bigvee_{i}\psi_{i}$ and $\bigwedge_{i}\psi_{i}$ denote respectively representatives of the classes $\bigvee_{i}[\psi_{i}]$ and $\bigwedge_{i}[\psi_{i}]$ (as described in [IKR1, §7]). Then

[TABLE]

We turn to the case where $\varphi:R\to U$ is a supervaluation, i.e., the supertropical monoid  $U$ is a semiring. We want to characterize the partial tangible lifts $\psi$ of $\varphi$ which are again supervaluations; in other terms, we want to determine the subset $[\varphi,\widetilde{\varphi}]\cap\operatorname{Cov}(v)$ of the interval $[\varphi,\widetilde{\varphi}]$ of $\operatorname{Cov}_{m}(v)$.

The set $Y(v)$ introduced at the end of [IKR4, §7] will play a decisive role. It consists of the products $ab\in R$ of elements $a,b\in R$ for which there exists some $a^{\prime}\in R$ with

[TABLE]

Henceforth, we call these products $ab$ the $v$-NC-products (in $R$). Let

[TABLE]

where $\mathfrak{q}$ is the support of $v$, $\mathfrak{q}=v^{-1}(0)$. As observed in [IKR4, §7], $\mathfrak{q}^{\prime}$ is an ideal of the monoid $(R,\cdot\;)$, while $\mathfrak{q}$ is an ideal of the semiring $R$.

Example 2.9**.**

Let $R$ be a supertropical semiring and let $\gamma:eR\to M$ be a semiring homomorphism to a bipotent semiring $M$. Then

[TABLE]

is a strict m-valuation. The $v$-NC-products are the products $yz$ with $y,z\in U$ such that there exists some $y^{\prime}\in R$ with

[TABLE]

Thus $Y(v)$ is the ideal $D_{0}(R,\gamma)$ of the supertropical semiring $R$, introduced in [IKR4, Definition 4.8].

Proposition 2.10**.**

If $\varphi$ is a supervaluation then $\varphi(\mathfrak{q}^{\prime})$ is contained in the ghost value set  $G(\varphi)$.

Proof.

We have seen in [IKR4, §7] that $\varphi(Y(v))\subset M$. Since $\varphi(\mathfrak{q})=\{0\}$, this implies that $\varphi(\mathfrak{q}^{\prime})\subset M\cap\varphi(R)=G(\varphi)$. ∎

Remark 2.11**.**

Here is a more direct argument that $\varphi(Y(v))\subset M$, than given in the proof of [IKR4, Theorem 7.12.i]. If $x\in Y(v)$, then we have $a^{\prime},a,b\in R$ with $x=ab$, $v(a^{\prime})<v(a)$, $v(a^{\prime}b)=v(ab)\neq 0$. Clearly $\varphi(x)=\varphi(a)\varphi(b)$ is an NC-product in the supertropical semiring  $U$ [IKR4, Definition 4.2], and thus $\varphi(x)$ is ghost, as already observed in [IKR4, Theorem 1.2].

Lemma 2.12**.**

Assume that $\varphi:R\to U$ is a surjective tangible m-supervaluation covering   $v$. Then $\varphi(R\setminus\mathfrak{q})=\mathcal{T}(U)$, $v(R)=M$, and $\varphi(Y(v))=S(U)$.444Recall that $S(U)$ denotes the set of tangible NC-products in $U$ [IKR4, Definition 4.2].

Proof.

a) We have $U=\varphi(R)\cup v(R)$, $\varphi(R\setminus\mathfrak{q})\subset\mathcal{T}(U)$, and $v(R)\subset M$. Since $U=\mathcal{T}(U)\dot{\cup}M$, this forces $\varphi(R\setminus\mathfrak{q})=\mathcal{T}(U)$ and $v(R)=M.$ b) Let $c\in Y(v)$. There exist $a,b,a^{\prime}\in R$ with $c=ab$, $v(a^{\prime})<v(a)$, $v(a^{\prime}b)=v(ab)\neq 0.$ It follows that $\varphi(c)=xy\neq 0$ with $x:=\varphi(a)$, $y:=\varphi(b)$, $v(a^{\prime})<ex$, $v(a^{\prime})y=exy.$ Thus $\varphi(c)$ is an NC-product in $U$. Moreover $\varphi(c)$ is tangible, hence $\varphi(c)\in S(U).$ Thus $\varphi(Y(v))\subset S(U).$ c) Let $x\in S(U)$ be given. Then $x=yz\in\mathcal{T}(U)$ with $y,z\in U$ and $y^{\prime}<ey$, $y^{\prime}z=eyz\neq 0$ for some $y^{\prime}\in M$. Clearly $y,z\in\mathcal{T}(U)$. We choose $a,b,a^{\prime}\in R$ with $\varphi(a)=y$, $\varphi(b)=z$, $v(a^{\prime})=y^{\prime}.$ Then $ey=v(a)$, $ez=v(b)$, and it follows that $v(a^{\prime})<v(a)$, $v(a^{\prime}b)=v(ab)\neq 0$. Thus $ab\in Y(v)$ and $x=\varphi(ab)$. This proves that $S(U)\subset\varphi(Y(v))$. ∎

Theorem 2.13**.**

Assume that $\varphi:R\to U$ is a supervaluation, i.e., $U$ is a semiring. Let $\widecheck{\varphi}$ denote the tangible lift of $\varphi$ outside the ideal $v(\mathfrak{q}^{\prime})=\{0\}\cup v(Y(v))$ of $M$,

[TABLE]

(cf. Construction 2.6).

1. (i)

$\widecheck{\varphi}$* is again a supervaluation. More precisely, $\widecheck{\varphi}$ coincides with the supervaluation $(\widetilde{\varphi})^{\wedge}$ associated to the tangible lift $\widetilde{\varphi}:R\to\widetilde{U}$ of $\varphi$ (cf. [IKR4, Definition 7.7]).* 2. (ii)

If $\psi$ is an m-supervaluation covering $v$ with $\varphi\leq\psi\leq\widetilde{\varphi}$, then $\psi$ is a supervaluation iff $\psi\leq\widecheck{\varphi}$. Thus

[TABLE]

Proof.

(i): $(\widetilde{\varphi})^{\wedge}$ is the map $\widetilde{\varphi}/E(\widetilde{U},S(\widetilde{U}))=\pi_{E(\widetilde{U},S(\widetilde{U}))}\circ\widetilde{\varphi}$ from $R$ to $\widecheck{U}:=\widetilde{U}/E(\widetilde{U},S(\widetilde{U})).$ Applying Lemma 2.12 to $\varphi$, we have

[TABLE]

where $s:\varphi(R)\to\mathcal{T}(\widetilde{U})_{0}$ is the tangible lifting map for $\varphi$. Moreover $\varphi(\mathfrak{q}^{\prime})=v(\mathfrak{q}^{\prime})$ by Proposition 2.10. Thus $\widecheck{U}=\widetilde{U}/E_{v(\mathfrak{q}^{\prime})}$ and $(\widetilde{\varphi})^{\wedge}=\widetilde{\varphi}/E_{v(\mathfrak{q}^{\prime})}=\widecheck{\varphi}.$ (ii): If $\psi$ is a supervaluation, then we know by Proposition 2.10 that $G(\psi)\supset v(\mathfrak{q}^{\prime})=G(\widecheck{\varphi})$, and hence by Theorem 2.4 that $\psi\leq\widecheck{\varphi}$. Conversely, if $\psi\leq\widecheck{\varphi}$, then $\psi$ is a supervaluation, since $\widecheck{\varphi}$ is a supervaluation (cf. [IKR4, Proposition 7.6]). ∎

Definition 2.14**.**

**

1. (i)

Given a supervaluation $\varphi:=R\to U$ covering $v$ we call

[TABLE]

the almost tangible lift of $\varphi$ (to a supervaluation) and we call $[\widecheck{\varphi}]\in\operatorname{Cov}(v)$ the almost tangible lift (in $\operatorname{Cov}(v)$) of the class $[\varphi]\in\operatorname{Cov}(v)$. 2. (ii)

If $\widecheck{\varphi}=\varphi$, we say that $\varphi$ itself is almost tangible.

Remarks 2.15**.**

**

1. (a)

Clearly $\varphi$ is almost tangible iff $G(\varphi)=v(\mathfrak{q}^{\prime})$. A subtle point here is that there may nevertheless exist elements $a\in R\setminus\mathfrak{q}^{\prime}$ with $\varphi(a)$ ghost. 2. (b)

If $\varphi$ is any supervaluation, then $\widecheck{\varphi}$ is almost tangible. 3. (c)

If $v$ happens to be a valuation, i.e., $M$ is cancellative, then $\widecheck{\varphi}=\widetilde{\varphi}$.

Proposition 2.16**.**

If $\psi$ is an almost tangible supervaluation dominating the supervaluation $\varphi$ (but not necessarily covering $v$), then $\psi$ dominates $\widecheck{\varphi}$.

Proof.

$\widetilde{\psi}\geq\widetilde{\varphi}$, and hence $\psi=(\widetilde{\psi})^{\wedge}\geq(\widetilde{\varphi})^{\wedge}=\widecheck{\varphi}.$ ∎

3. Tangible and mixing transmissions

The intent of this section is to display any transmission $\alpha:U\to V$ of supertropical monoids as a product $\alpha=\alpha_{m}\circ\alpha_{t}$ of a tangible transmission $\alpha_{t}$ and a “mixing” transmission $\alpha_{m}$, and to study properties of these factors. Tangible transmissions were introduced in [IKR4, Definition 2.1], while mixing transmissions are introduced below. A key result for this factorization is provided by Theorem 3.2 on TE-relations [IKR4, Definition 1.7].

Definition 3.1**.**

A TE-relation $E$ on a supertropical monoid $U$ is called ghost separating, if no element $x\in\mathcal{T}(U)$ with $x\not\sim_{E}0$ is $E$-equivalent to an element $y\in\mathcal{G}(U)$. In other terms, $E$ is ghost separating iff $\pi_{E}:U\to U/E$ is a tangible transmission [IKR4, Definition 2.3].

Theorem 3.2**.**

Let $E$ be a TE-relation on a supertropical monoid $U$. Define the binary relation ${\widetilde{E}}$ on $U$ by

[TABLE]

1. (i)

${\widetilde{E}}$* is a TE-relation on $U$ with ${\widetilde{E}}\subset E$ and ${\widetilde{E}}|M=E|M$.* 2. (ii)

${\widetilde{E}}$* is ghost separating.* 3. (iii)

If $F$ is a ghost separating TE-relation on $U$ with $F\subset E$, then $F\subset{\widetilde{E}}$.

Proof.

(i): It is obvious that ${\widetilde{E}}$ is a multiplicative equivalence relation on $U$ and ${\widetilde{E}}|M=E|M$. Assume that $x\in U$ and $ex\sim_{\widetilde{E}}0$. Then $ex\sim_{E}0$, hence $x\sim_{E}0$, implying that $x\sim_{\widetilde{E}}0$. Thus ${\widetilde{E}}$ is a TE-relation. Clearly ${\widetilde{E}}\subset E$. (ii): Assume that $x\in\mathcal{T}(U)$, $y\in\mathcal{G}(U)$, but $x\not\sim_{E}0$ and $y\not\sim_{E}0$. Then, $x$ and $y$ are not ${\widetilde{E}}$-equivalent, since the second condition in $(*)$ fails for $z=1$. Thus ${\widetilde{E}}$ is ghost separating. (iii): Assume that $x,y\in U$ and $x\sim_{F}y$. Then $x\sim_{E}y$. Given $z\in U$, since $F$ is ghost separating, either $xz,yz\in\mathcal{T}(U)$ or $xz,yz\in\mathcal{G}(U)$, or $xz\in[0]_{F}\subset[0]_{E}$ 555Recall that $[0]_{F}$ denotes the $F$-equivalence class of $0\in U$.. Thus $x\sim_{\widetilde{E}}y$. This proves that $F\subset{\widetilde{E}}$. ∎

Definition 3.3**.**

**

1. (a)

We call ${\widetilde{E}}$ the ghost separating refinement of the TE-relation $E$. 2. (b)

If $E$ is an MFCE-relation, we alternatively say that ${\widetilde{E}}$ is the tangible refinement of $E$, to be compatible with the definition of a tangible MFCE-relation in **[IKR4, Definition 2.4]**.

In the second condition on the right side of $(*)$, defining ${\widetilde{E}}$, we may discard the elements  $z$ of $eU$, since then we have $xz,yz\in eU$ for free. We may also use the sets $\mathcal{T}(U)$, $\mathcal{G}(U)$, $[0]_{E}$ instead of $\mathcal{T}(U)\cup\{0\}$, $eU$, $[0]_{E}$, although $\mathcal{T}(U)$ or $\mathcal{G}(U)$ may not be closed under multiplication. This leads to the following description of ${\widetilde{E}}$, which perhaps looks more natural than $(*)$.

Remark 3.4**.**

Let $x,y\in U$. The following are equivalent.

1. (i)

$x\sim_{{\widetilde{E}}}y$. 2. (ii)

$x\sim_{E}y$, and for every $z\in\mathcal{T}(U)$ the elements $xz,yz$ are both contained in one of the sets $\mathcal{T}(U),\mathcal{G}(U),[0]_{E}$.

The description of ${\widetilde{E}}$ becomes much simpler if $U$ is unfolded, i.e., $\mathcal{T}(U)_{0}$ is closed under multiplication.

Remark 3.5**.**

Assume that $E$ is a TE-relation on an unfolded supertropical monoid $U$. Let $x,y\in U$. Then $x\sim_{{\widetilde{E}}}y$ iff $x\sim_{E}y$ and the elements $x,y$ are both contained in one of the sets $\mathcal{T}(U)$, $\mathcal{G}(U)$, $[0]_{E}$. Indeed, now the last property is inherited by the pair $xz,yz$ from the pair $x,y$ for any $z\in U$.

We are ready for introducing the class of “mixing” transmissions. It will be convenient to use the following catch-phrases: We say that two elements $x,y$ of $U\setminus\{0\}$ are of different kind, if either $x\in\mathcal{T}(U)$, $y\in\mathcal{G}(U)$, or $x\in\mathcal{G}(U)$, $y\in\mathcal{T}(U)$. Otherwise we say that $x,y$ are of the same kind.

Definition 3.6**.**

Let $\alpha:U\to V$ be a transmission between supertropical monoids. We say that $\alpha$ is mixing, if $\alpha^{-1}(0)=\{0\}$, and if for any two different elements $x,y$ of $U\setminus\{0\}$ with $\alpha(x)=\alpha(y)$ there exists some $z\in U$ with $\alpha(xz)\neq 0$ (hence also $\alpha(yz)\neq 0$) and $xz$, $yz$ of different kind.

We start the study of this new class of transmissions with some simple observations.

Remarks 3.7**.**

**

1. (i)

If $\alpha:U\to V$ is a transmission with $\alpha^{-1}(0)=\{0\}$, and $E:=E(\alpha)$ is the equivalence relation determined by $\alpha$ ($x\sim_{E}y$* iff $\alpha(x)=\alpha(y)$), then $\alpha$ is mixing iff the ghost separating refinement ${\widetilde{E}}$ of $E$ is trivial, i.e., $x\sim_{\widetilde{E}}y$ implies $x=y$. This is obvious from the definition of ${\widetilde{E}}$ (cf. Remark 3.4).* 2. (ii)

If $\alpha:U\to V$ is mixing, then the ghost part $\alpha^{\nu}:eU\to eV$ of $\alpha$ is injective. Indeed, if $x,y\in eU$ and $\alpha(x)=\alpha(y)$, then $xz,yz\in eV$ for every $z\in U$. This forces $x=y$. Thus every surjective mixing transmission is a fiber contraction. 3. (iii)

Let $\alpha_{1}:U\to U_{1}$ and $\alpha_{2}:U_{1}\to U_{2}$ be transmissions, and assume that $\alpha_{2}$ is injective. Then $\alpha_{2}\circ\alpha_{1}$ is mixing iff $\alpha_{1}$ is mixing. Every mixing transmission $\alpha:U\to V$ is the composite $i\circ\alpha_{0}$ of a surjective mixing transmission $\alpha_{0}:U\to\alpha(U)$ and an inclusion map $i:\alpha(U)\hookrightarrow V$. 4. (iv)

If $\alpha:U\to V$ is mixing and $j:U^{\prime}\to U$ is an injective transmission, then $\alpha\circ j:U^{\prime}\to V$ is again mixing.

Lemma 3.8**.**

Assume that $\alpha:U\to V$ and $\beta:V\to W$ are transmissions and that $\beta\alpha:U\to W$ is mixing. Then $\alpha$ is mixing.

Proof.

Since $(\beta\alpha)^{-1}(0)=\{0\}$, also $\alpha^{-1}(0)=\{0\}$. Let $x,y$ be different elements in $U\setminus\{0\}$ with $\alpha(x)=\alpha(y)$. Then $\beta\alpha(x)=\beta\alpha(y)$. Since $\beta\alpha$ is mixing, there exists some $z\in U$ with $\beta\alpha(xz)\neq 0$, hence $\alpha(xz)\neq 0$, and $xz$, $yz$ of different kind. This proves that $\alpha$ is mixing. ∎

Lemma 3.9**.**

Assume that $\alpha:U\to V$ is a surjective transmission which is both tangible and mixing, Then $\alpha$ is an isomorphism.

Proof.

A bijective transmission is an isomorphism. Thus it suffices to verify that $\alpha$ is injective. Let $x,y$ be different elements of $U$. Suppose that $\alpha(x)=\alpha(y)$. Since $\alpha^{-1}(0)=\{0\}$, both $x,y$ are not zero. Since $\alpha$ is mixing, we conclude that there exists some $z\in U$ with $\alpha(xz)\neq 0$, $\alpha(yz)\neq 0$, and, say $xz\in\mathcal{T}(U)$, $yz\in\mathcal{G}(U)$. Since $\alpha$ is tangible, it follows that $\alpha(xz)\in\mathcal{T}(V)\cup\{0\}$, $\alpha(yz)\in eV$. But $\alpha(xz)=\alpha(yz)$, which forces $\alpha(xz)=0$, a contradiction. This proves that $\alpha$ is injective. ∎

We are ready for the factorization theorem, announced at the beginning of the section.

Theorem 3.10**.**

Let $\alpha:U^{\prime}\twoheadrightarrow U$ be a surjective transmission, $M^{\prime}:=eU^{\prime}$, $M:=eU$, and

[TABLE]

1. (i)

There exists a commuting triangle

[TABLE]

where $\alpha_{t}$ is a surjective tangible transmission covering $\gamma$ (in particular $eV=M$) and $\alpha_{m}$ is a mixing transmission over $M$. 2. (ii)

Given a factorization $\alpha=\alpha_{m}\alpha_{t}$ as described in (i), the following holds. If

[TABLE]

is a factorization of $\alpha$ with $\beta$ surjective and tangible and $\mu$ a fiber contraction over $M$ (in particular $eW=M$), then there exists a unique fiber contraction $\zeta:W\to V$ over $M$ such that $\zeta\beta=\alpha_{t}$, which forces $\alpha_{m}\zeta=\mu$. If in addition $\mu$ is mixing, then  $\zeta$ is an isomorphism.

We indicate this by the following diagram:

[TABLE]

Proof.

a) Let $E:=E(\alpha)$, and take

[TABLE]

The transmission $\alpha_{t}$ is tangible and covers $\gamma$ (cf. Theorem 3.2). Since ${\widetilde{E}}\subset E$, there is a unique map $\alpha_{m}:V\to U$ with $\alpha=\alpha_{m}\alpha_{t}$. Since $\alpha$ and $\alpha_{t}$ are surjective transmissions, also $\alpha_{m}$ is a surjective transmission. Since both $\alpha$ and $\alpha_{t}$ cover $\gamma$, we have $\alpha^{\nu}_{m}=\operatorname{id}_{M}$. Thus  $\alpha_{m}$ is a fiber contraction over $M$.

We verify that $\alpha_{m}$ is mixing. Let $x,y\in U^{\prime}$ be given. Let $\bar{x}:=[x]_{\widetilde{E}}$, $\bar{y}:=[y]_{\widetilde{E}}$ and assume that $\bar{x}\neq\bar{y}$, i.e., $x\not\sim_{\widetilde{E}}y$. Assume also that $\alpha_{m}(\bar{x})=\alpha_{m}(\bar{y})$, i.e., $x\sim_{E}y$, in other terms, $\alpha(x)=\alpha(y)$. By Remark 3.4 we conclude that there exists some $z\in U$ with $\alpha(xz)\neq 0$, $\alpha(yz)\neq 0$, and $xz,yz$ of different kind. Then, with $\bar{z}:=[z]_{\widetilde{E}}$, we have $\alpha_{m}(\bar{x}\bar{z})\neq 0$, $\alpha_{m}(\bar{y}\bar{z})\neq 0$, and $\bar{x}\bar{z},\bar{y}\bar{z}$ of different kind, since ${\widetilde{E}}$ is ghost separating. This proves that $\alpha_{m}$ is mixing. b) We continue with the factorization $\alpha=\alpha_{m}\alpha_{t}$ as just constructed. Let $\alpha=\mu\beta$ be a factorization of $\alpha$ with $\beta:U^{\prime}\to W$ tangible and surjective, and $\mu:W\to U$ a fiber contraction over  $M$. Since $\mu^{\nu}=\operatorname{id}_{M}$, this forces $\beta^{\nu}=\gamma$. We have $E(\beta)\subset E(\alpha)=E$, and $E(\beta)$ is ghost separating. Thus $E(\beta)\subset{\widetilde{E}}$. We conclude that there exists a surjective transmission $\zeta:W\to U^{\prime}/{\widetilde{E}}=V$ with $\zeta\beta=\alpha_{t}$. Since $\beta^{\nu}=\alpha_{t}^{\nu}=\gamma$, it follows that $\zeta^{\nu}=\operatorname{id}_{M}$, i.e., $\zeta$ is a fiber contraction over $M$, and thus $\alpha_{m}\alpha_{t}=\alpha_{m}\zeta\beta=\mu\beta$. Since $\beta$ is surjective, this implies $\mu=\alpha_{m}\zeta$. c) Assume in addition that $\mu$ is mixing. Since $\alpha_{t}=\zeta\beta$ and $\beta$ are tangible, also $\zeta$ is tangible. Since $\mu=\alpha_{m}\zeta$ is mixing, if follows by Lemma 3.8 that $\zeta$ is mixing, and then by Lemma 3.9 that $\zeta$ is an isomorphism. d) It is now evident that the statement (ii) of the theorem holds for any factorization $\alpha=\alpha_{m}\alpha_{t}$ as described in part (i), and not only for the one constructed in step a) of the proof. ∎

Remark 3.11**.**

Assume that in Diagram (3.1) the monoid $U^{\prime}$ is a (supertropical) semiring. Then also $V$ is a semiring, cf. [IKR4, Theorem 1.6.(ii)], and consequently $U$ is a semiring. Thus the whole Diagram (3.1) is situated in the category of supertropical semirings. In particular the mixing property of $\alpha_{m}$ takes place in this category.

Theorem 3.10 can be readily generalized to a factorization theorem of a transmission which is not necessarily surjective.

Corollary 3.12**.**

Assume that $\alpha:U^{\prime}\to U$ is any transmission. Let $\gamma:=\alpha^{\nu}:M^{\prime}\to M$ denote the ghost part of $\alpha$.

1. (i)

There exists a surjective tangible transmission $\alpha_{t}:U^{\prime}\to V$ and a mixing transmission $\alpha_{m}:V\to U$ covering the injection $i:\gamma(M)\hookrightarrow M^{\prime}$ such that $\alpha=\alpha_{m}\circ\alpha_{t}$. 2. (ii)

Given a second factorization $\alpha=\mu\circ\beta$ with $\beta:U^{\prime}\to W$ surjective and tangible, and $\mu:W\to U$ mixing, there exists a unique isomorphism \zeta:W\xrightarrow{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}V such that $\zeta\beta=\alpha_{t}$ and $\alpha_{m}\zeta=\mu.$

Proof.

(i): Let $\alpha_{0}:U^{\prime}\to\alpha(U^{\prime})$ denote the transmission obtained from $\alpha$ by restricting the target $U$ to $\alpha(U^{\prime})$. We have a factorization $\alpha_{0}=(\alpha_{0})_{m}\circ(\alpha_{0})_{t}$ with the properties stated in Theorem 3.10. Let $\alpha_{t}:=(\alpha_{0})_{t}$ and $\alpha_{m}:=j\circ(\alpha_{0})_{m}$ with $j$ the inclusion map from $\alpha(U^{\prime})$ to $U$. Then $\alpha=\alpha_{m}\circ\alpha_{t}$. Here $\alpha_{t}$ is surjective and tangible, while $\alpha_{m}$ is mixing, as follows form Remak 3.7.(iv). (ii): By Theorem 3.10.(ii) there is a unique isomorphism $\zeta:W\to\alpha_{0}(U)$ with $\zeta\beta=(\alpha_{0})_{t}=\alpha_{t}$. Since $\alpha=\alpha_{m}\alpha_{t}=\alpha_{m}\zeta\beta=\mu\beta$, and $\beta$ is surjective, this forces $\alpha_{m}\zeta=\mu$. ∎

Definition 3.13**.**

We call $\alpha_{t}$ “the” surjective tangible part and $\alpha_{m}$ the mixing part of the transmission $\alpha$. $\alpha=\alpha_{m}\circ\alpha_{t}$ is called the $(t,m)$-factorization of $\alpha.$ When $\alpha$ is surjective, we call $\alpha_{t}$ briefly the tangible part of $\alpha$.

Remark 3.14**.**

Let $\alpha:U\to V$ be any transmission with ghost part $\gamma:=\alpha^{\nu}:M\to N$. By [IKR4, Theorem 1.13] we have a unique factorization

[TABLE]

where $\beta$ is a fiber contraction over $N=eV$. Since $\pi_{F(U,\gamma)}$ is tangible and surjective, it is obvious that

[TABLE]

The remark indicates that finding $(t,m)$-factorizations is essentially an issue on fiber contractions, and thus can be turned into a problem about MFCE-relations.

From part a) of the proof of Theorem 3.10 now the following is clear.

Scholium 3.15**.**

Let $E$ be an MFCE-relation on a supertropical monoid $U$. Then the transmission $\pi_{E}:U\to U/E$ has the tangible part $(\pi_{E})_{t}=\pi_{{\widetilde{E}}}$ and the mixing part $(\pi_{E})_{m}=\pi_{E/{\widetilde{E}}}$. Here $E/{\widetilde{E}}$ denotes the equivalence relation induced by $E$ on $U/{\widetilde{E}}$, i.e., for  $x,y\in U$

[TABLE]

Example 3.16** (The relation ${\widetilde{E}}(\nu)$).**

Let $U$ be a supertropical monoid and $M:=eU$. For any subset $L$ of $U$ and $x\in U$ we define

[TABLE]

The equivalence relation $E(\nu):=E(\nu_{U})$ induced by the ghost map $\nu_{U}$ clearly is the coarsest MFCE-relation on $U$. Notice that $x\sim_{E(\nu)}y$ iff $ex=ey$. Theorem 3.2 tells us that the relation ${\widetilde{E}}(\nu):=(E(\nu))^{\sim}$ is the coarsest ghost separating MFCE-relation on $U$. Furthermore, by that theorem two elements $x,y$ of $U$ are ${\widetilde{E}}(\nu)$-equivalent iff $ex=ey$ and for every $z\in U$ both $xz$ and $yz$ lie in $\mathcal{T}(U)_{0}$ or in $\mathcal{G}(U)_{0}$. This can also be phrased as follows.

[TABLE]

Remark 3.17**.**

If $E$ is any MFCE-relation, we read off from the preceding remark and Theorem 3.2 that ${\widetilde{E}}=E\cap{\widetilde{E}}(\nu)$. Since any equivalence relation finer than ${\widetilde{E}}(\nu)$ is ghost separating, it is clear form Remark 3.14 that $E\cap{\widetilde{E}}(\nu)$ is the coarsest ghost separating MFCE-relation finer than $E$.

Example 3.18** (The relation $E_{t}$).**

We return to the fiber conserving equivalence relation $E_{t}:=E_{t,U}$ mentioned near [IKR4, Definition 2.3]. For $x,y\in U$

[TABLE]

Clearly ${\widetilde{E}}(\nu)\subset E_{t}$. If $E_{t}$ is multiplicative, then $E_{t}$ is MFCE and ghost separating, hence $E_{t}\subset{\widetilde{E}}(\nu).$ We conclude that ${\widetilde{E}}(\nu)=E_{t}$ iff for any elements $x,y,z\in\mathcal{T}(U)$ with $ex=ey$ the elements $xz,yz$ are both tangible or equal. Certainly this happens if $\mathcal{T}(U)\cup\{0\}$ is closed under multiplication, but there are other cases.

As in [IKR4, §4] (and for $U$ a semiring ordering in [IKR1, §7]) we denote the poset consisting of all MFCE-relations on $U$ by $\operatorname{MFC}(U).$ Recall that $\operatorname{MFC}(U)$ is a complete lattice with top element $E(\nu_{U})$ and bottom element $\operatorname{diag}(U)$, cf. [IKR1, §7]. We take a brief look at the sets of those $E\in\operatorname{MFC}(U)$ for which the associated fiber contraction $\pi_{E}:U\to U/E$ is either tangible or mixing.

Definition 3.19**.**

**

1. (a)

Let $E\in\operatorname{MFC}(U)$. We say that $E$ is tangible (resp. mixing) if the transmission  $\pi_{E}$ is tangible (resp. mixing). 2. (b)

We denote the set of all tangible $E\in\operatorname{MFC}(U)$ by $\operatorname{MFC}_{{\operatorname{t}}}(U)$, and the set of all mixing $E\in\operatorname{MFC}(U)$ by $\operatorname{Mix}(U).$ We regard both $\operatorname{MFC}_{{\operatorname{t}}}(U)$ and $\operatorname{Mix}(U)$ as subposets of the lattice $\operatorname{MFC}(U).$

N.B. An MFCE-relation $E$ on $U$ is tangible iff the TE-relation $E$ is ghost separating. Tangible MFCE-relations had already been introduced in [IKR4, Definition 2.4].

Scholium 3.20**.**

It follows from Remark 3.17, that $\operatorname{MFC}_{{\operatorname{t}}}(U)$ is the set of all $E\in\operatorname{MFC}(U)$ with $E\subset{\widetilde{E}}(\nu)$, while $\operatorname{Mix}(U)$ is the set of all $E\subset\operatorname{MFC}(U)$ with $E\cap{\widetilde{E}}(\nu)=\{\operatorname{diag}U\}$. Thus both $\operatorname{MFC}_{t}(U)$ and $\operatorname{Mix}(U)$ are lower subsets of $\operatorname{MFC}(U)$, and

[TABLE]

Moreover, $\operatorname{MFC}_{{\operatorname{t}}}(U)$ is a complete sublattice of $\operatorname{MFC}(U).$

Lemma 3.21**.**

Assume that $(E_{i}\ {|}\ i\in I)$ is a family of MFCE-relations on $U$ with the property that for any two indices $i,j\in I$ there exists some $k\in I$ with $E_{i}\subset E_{k}$ and $E_{j}\subset E_{k}$. Let $E:=\bigcup_{i\in I}E_{i}\subset U\times U.$

1. (i)

$E$* is again an MFCE-relation on $U$. Thus*

[TABLE] 2. (ii)

For any $L\in\operatorname{MFC}(U)$ we have

[TABLE] 3. (iii)

In particular $(L={\widetilde{E}}(\nu))$

[TABLE]

Proof.

(i): An easy check starting from the definition of an MFCE-relation. (First verify that  $E$ is an equivalence relation.) (ii): Apply (i) to the family ($E_{i}\cap L\ {|}\ i\in I$). (iii): Now clear, since for any $F\in\operatorname{MFC}(U)$ we have ${\widetilde{F}}=F\cap{\widetilde{E}}(\nu)$ (cf. Remark 3.17). ∎

Theorem 3.22**.**

For every $E\in\operatorname{Mix}(U)$ there exists a maximal element $F$ of $\operatorname{Mix}(U)$ with  $E\subset F$.

Proof.

Let $(E^{\prime}_{i}\ {|}\ i\in I)$ be a chain in the poset $\operatorname{Mix}(U)$ with $E\subset E^{\prime}_{i}$ for all $i\in I$. Then $E^{\prime}:=\bigcup_{i}E^{\prime}_{i}$ is a mixing MFCE-relation on $U$ by Lemma 3.21.(iii). The claim of the theorem follows by Zorn’s lemma. ∎

$\operatorname{Mix}(U)$ will be a sublattice of $\operatorname{MFC}(U)$ only in rather degenerate cases.

4. The $(t,m)$-factorization of the composite of two transmissions

Given two transmissions $\alpha:U\to V$ and $\beta:V\to W$ with $(t,m)$-factorizations $\alpha=\alpha_{m}\circ\alpha_{t}$, $\beta=\beta_{m}\circ\beta_{t}$, we intend to determine “the” $(t,m)$-factorization of the composite $\beta\circ\alpha:U\to W$.

First, we have the obvious fact that the composite of two tangible transmissions is again tangible. This is also the case for mixing transmissions.

Proposition 4.1**.**

If $\alpha:U\to V$ and $\beta:V\to W$ are mixing transmissions, then $\beta\circ\alpha:U\to W$ is mixing as well.

Proof.

1. $(\beta\alpha)^{-1}=\{0_{U}\}$, since $\beta^{-1}(0_{W})=\{0_{V}\}$ and $\alpha^{-1}(O_{V})=\{0_{U}\}$.
2. We now verify the claim in the case that $\alpha$ is surjective. Given $x,y\in U\setminus\{0\}$, where $x\neq y$ and $\beta\alpha(x)=\beta\alpha(y)$, we need to find some $z\in U$, necessarily $\neq 0$, such that $\beta\alpha(xz)\neq 0$ and $xz$ and $yx$ are of different kind.

Assume first that $\alpha(x)=\alpha(y)$. Since $\alpha$ is mixing, there exists $z\in U$ for which $xz\neq 0$, $yz\neq 0$ and $xz$, $yz$ are of different kind. There remain the case that $\alpha(x)\neq\alpha(y)$. By Step  1 it is clear that both $\alpha(x)$ and $\alpha(y)$ are not zero (and also $\beta\alpha(x)=\beta\alpha(y)\neq 0$). Since  $\beta$ is mixing, there is some $v\in V\setminus\{0\}$ with $\alpha(x)v$ and $\alpha(y)v$ of different kind. As $\alpha$ is surjective, there is some $z\in U\setminus\{0\}$ with $\alpha(z)=v$. Then $\alpha(x)v=\alpha(xz)$, $\alpha(y)v=\alpha(yz)$. Since $\alpha(x)v$ and $\alpha(y)v$ are of different kind, clearly also $xz$ and $yz$ are of different kind. 3) When $\alpha$ is not surjective, we have a factorization $\alpha=j\alpha_{0}$ where $\alpha_{0}:U\twoheadrightarrow\alpha(U)$ is a surjective transmission and $j:\alpha(U)\hookrightarrow V$ is an injective transmission. Then $\alpha=(\beta j)\circ\alpha_{0}$. We conclude by Lemma 3.8 that $\alpha_{0}$ is mixing, since $j\alpha_{0}$ is mixing, and by Remark 3.4.(iv) that $\beta j$ is mixing. Finally, by Step 2 we obtain that $\beta\alpha=(\beta j)\circ\alpha_{0}$ is mixing. ∎

Theorem 4.2**.**

Assume that $\alpha:U\to U^{\prime}$ and $\beta:U^{\prime}\to U^{\prime\prime}$ are transmissions with $(t,m)$-factorizations $\alpha=\alpha_{m}\circ\alpha_{t}$, $\beta=\beta_{m}\circ\beta_{t}$, having the surjective tangible parts $\alpha_{t}:U\to~{}V^{\prime}$, $\beta_{t}:U^{\prime}\twoheadrightarrow V^{\prime\prime}$ and the mixing parts $\alpha_{m}:V^{\prime}\to U^{\prime}$, $\beta_{m}:V^{\prime\prime}\twoheadrightarrow U^{\prime\prime}$, respectively (cf. Definition  3.13). Consider also the $(t,m)$-factorization of $\rho:=\beta_{t}\circ\alpha_{m}:V^{\prime}\to V^{\prime\prime}$ with surjective tangible part $\rho_{t}:V^{\prime}\to W$ and mixing part $\rho_{m}:W\to V^{\prime\prime}$, so that all together the following commutative diagram of transmissions appears:

[TABLE]

Then the transmission $\beta\alpha:U\to U^{\prime\prime}$ has the surjective tangible part $\rho_{t}\alpha_{t}$ and the mixing part $\beta_{m}\rho_{m}$.

Proof.

We have the factorization

[TABLE]

The transmission $\rho_{t}\alpha_{t}$ is surjective tangible, while $\beta_{m}\rho_{m}$ is mixing by Proposition 4.1. The assertion follows from the uniqueness of $(t,m)$-factorizations, stated in Corollary  3.12.(ii). ∎

5. Ideal-generating sections and tyrants

We exhibit a class of special elements of the poset $\operatorname{Mix}(U)$, introduced in §3.

Definition 5.1**.**

Let $U$ be a supertropical monoid with $M:=eU.$ A map $s:M\to U$ is called an ig-section (= ideal-generating section) in $U$, if

[TABLE]

Remark 5.2**.**

Assuming $({\operatorname{S1}})$, the condition $({\operatorname{S2}})$ is equivalent to:

[TABLE]

Proof.

$({\operatorname{S2}})\Rightarrow({\operatorname{S2}}^{\prime})$: A priori we have $es(x)y=xy\in M$. By $({\operatorname{S2}})$ we know that either $s(x)y=xy$ or $s(x)y=s(xy).$ For $s(x)y\in\mathcal{T}(U)$, this implies $s(x)y=s(xy).$ $({\operatorname{S2}}^{\prime})\Rightarrow({\operatorname{S2}})$: Given $x\in M$, $y\in U$, we verify that $s(x)y\in M\cup s(M)$. If $s(x)y\notin M$, then $({\operatorname{S2}}^{\prime})$ states that $s(x)y=s(xy)\in s(M)$. ∎

Theorem 5.3**.**

**

1. (i)

If $s$ is an ig-section in $U$, then the ideal compression (cf. **[IKR4, Definition 2.5]**)

[TABLE]

is an MFCE-relation such that each $E$-equivalence class contains at most one tangible element, hence consists of at most 2 elements. More precisely, the only equivalence classes, which are not one-element sets, are classes of the form $[a]_{E}=\{a,s(a)\}$ with $a\in M$, $s(a)\in\mathcal{T}(U).$ 2. (ii)

All these MFCE-relations $E(s)$ are mixing. 3. (iii)

We have a bijection $s\mapsto E(s)$ from the set $\operatorname{IGS}(U)$ of all ig-sections in $U$ onto the set of all MFCE-relations $E$ on $U$ with $|[x]_{E}\cap\mathcal{T}(U)|\leq 1$ for every $x\in U.$

Proof.

(i): Given an ig-section $s:M\to U,$ let $x,y\in U$, $x\neq y$ and $x\sim_{E(s)}y.$ Then $ex=ey=:a$ and $x\sim_{E(s)}a\sim_{E(s)}y.$ Since $\mathfrak{A}(s)\cap U_{a}=\{a,s(a)\}$, this forces $x=a$, $y=s(a)\in\mathcal{T}(U)$, or vice versa. Thus $[x]_{E(s)}=[a]_{E(s)}=\{a,s(a)\}$. (ii): Now obvious. (iii): Let $E$ be an MFCE-relation on $U$ such that each equivalence class contains at most one tangible element. Given $a\in M$, we have

[TABLE]

where either $s(a)=a$ or $s(a)$ is the unique tangible element in $[a]_{E}$. This gives a function $s:M\to U$ with $es(a)=a$ for all $a\in M$. Suppose $y\in U$ and $s(a)y\in\mathcal{T}(U)$. Then $s(a)\in\mathcal{T}(U)$ and $a\sim_{E}s(a)$. Since $E$ is multiplicative, it follows that $ay\sim_{E}s(a)y\in\mathcal{T}(U)$, hence $s(a)y=s(ay)$. Claim (iii) is now obvious. ∎

Definition 5.4**.**

We carry over the partial order on the subposet $\{E(s)\ {|}\ s\in\operatorname{IGS}(U)\}$ of the lattice $\operatorname{MFC}(U)$ of MFCE-relations on $U$ to the set $\operatorname{IGS}(U)$ by the bijection $s\mapsto E(s).$ In other terms, for $s,t\in\operatorname{IGS}(U)$ we define

[TABLE]

Clearly the following holds.

Remark 5.5**.**

Let $s,t\in\operatorname{IGS}(U)$. The following are equivalent.

1. (i)

$t\leq s.$** 2. (ii)

For any $a\in M$ either $t(a)=s(a)$ or $t(a)=a$.

By Theorem 5.3 it is obvious that $\{E(s)\ {|}\ s\in\operatorname{IGS}(U)\}$ is a lower set of the lattice $\operatorname{MFC}(U)$ contained in $\operatorname{Mix}(U)$, and that the trivial ig-section $a\mapsto a$ is the bottom element of $\operatorname{IGS}(U).$ Moreover we clearly have the following

Proposition 5.6**.**

The infimum $s\wedge t$ of $s$ and $t$ exists in the poset $\operatorname{IGS}(U)$ and is given  by

[TABLE]

Moreover $E(s\wedge t)=E(s)\cap E(t).$

In general, two elements $s,t$ of the poset $\operatorname{IGS}(U)$ need not have a common upper bound, but if they do, there exists a least upper bound (= supremum) of $s,t$. More precisely, we have the following fact.

Proposition 5.7**.**

Assume that $s,t$ are ig-sections in $U$, and that there exists an ig-section $u$ in $U$ with $s\leq u$, $t\leq u$. Then the function $s\vee t:M\to U$ given by ($a\in M$)

[TABLE]

is a well defined ig-relation in $U$, and is the supremum of $s$ and $t$ in the poset $\operatorname{IGS}(U).$ In the lattice $\operatorname{MFC}(U)$

[TABLE]

Proof.

Recall that for every $a\in M$ both $s(a)$ and $t(a)$ are elements of the set $\{a,u(a)\}$. Thus the function $s\vee t$ is certainly well defined (and independent of the choice of $u$). It is routine to verify that $s\vee t$ is an ig-section and is the supermum of $s,t$ in $\operatorname{IGS}(U)$. The isomorphism $s\mapsto E(s)$ from $\operatorname{IGS}(U)$ onto a lower subset of the lattice $\operatorname{MFC}(U)$ implies that $E(s)\vee E(t)=E(s\vee t).$ ∎

By the same vein we obtain.

Proposition 5.8**.**

Assume that $(s_{i}\ {|}\ i\in I)$ is a family in the poset $\operatorname{IGS}(U)$ such that for any two indices $i,j\in I$ the elements $s_{i},s_{j}$ have an upper bound in $\operatorname{IGS}(U)$ (hence the supremum $s_{i}\vee s_{j}$ exists by Proposition 5.7). Then we have a well defined function $s:M\to U$ with $s(a):=s_{i}(a)$ if there exists some $i\in I$ with $s_{i}(a)\in\mathcal{T}(U)$, and $s(a):=a$ else. The function $s$ is an ig-section in $U$, and is the supermum of the family $(s_{i}\ {|}\ i\in I)$ in $\operatorname{IGS}(U)$, $s=\bigvee_{i}s_{i}$. In the complete lattice $\operatorname{MFC}(U)$ we have

[TABLE]

Applying this proposition to chains in $\operatorname{IGS}(U)$ we obtain by Zorn’s lemma

Corollary 5.9**.**

For every $s\in\operatorname{IGS}(U)$ there exists a maximal element $s^{\prime}$ of the poset $\operatorname{IGS}(U)$ with $s\leq s^{\prime}$.

Definition 5.10**.**

We call an ig-section $s:M\to U$ primitive, if there exists some $x\in\mathcal{T}(U)$ such that the ideal $\mathfrak{A}(s)=M\cup s(M)$ of $U$ is generated by $e$ and $x$, $\mathfrak{A}(s)=eU\cup xU$. In this case we call $x$ a generator of the section $s$.

In more elaborate term this means that $x=s(a)\in\mathcal{T}(U)$ with $a:=ex$, and if $z:=s(c)\in\mathcal{T}(U)$ for some $c\in M$, then there exists some $y\in U$ such that $z=xy$. As a consequence, all products $xy$ with $y\in U$, $xy\in\mathcal{T}(U)$, $exy=c$, are equal. Indeed, we have $xy=s(a)y=s(ay),$ hence $c=es(ay)=ay$, an thus $xy=s(c).$

To get a better grasp at this situation we introduce some terminology concerning divisibility in the subset $\mathcal{T}(U)$ of the monoid $U$.

Definition 5.11**.**

Given $x,y\in\mathcal{T}(U)$ we say that $z$ is a son of $x$ and $x$ is a father of $z$, if $z=xy$ with some $y\in U$ (which then necessarily lies in $\mathcal{T}(U)$). An element $x$ of $\mathcal{T}(U)$ is said to be a tyrant of $U$ (or “$a$ is tyrannic in $U$”), if for any $c\in M$ there exists at most one son $z$ of $x$ with $ez=c.$

The term “tyrant” alludes to the property of $x$ that every tangible “genetic outcome”  $xy$ of pairing $x$ with some $y\in\mathcal{T}(U)$ is completely determined by its ghost $exy$ and  $x$. Notice that the ghost $b:=ey$ is not uniquely determined by $a:=ex$ and $c$, since $M$ may not be cancellative.

In this terminology we can recast Definition 5.10 as follows.

Remark 5.12**.**

A function $s:M\to U$ with $\nu_{U}\circ s=\operatorname{id}_{M}$ is a primitive ig-section iff there exists a tyrant $x$ in $U$ such that $s(M)\cap\mathcal{T}(U)$ is the set of all sons of $x$. These tyrants $x$ are then the generators of the primitive ig-section $s$.

A primitive ig-section may have several generators, but they all are “associated” in the following sense.

Definition 5.13**.**

We call two elements $x,z$ of $\mathcal{T}(U)$ associated (in $\mathcal{T}(U)$) if $z$ is a son of  $x$ and $x$ is a son of $z.$ We then write $x\sim_{\mathcal{T}}z$ (or more precisely $x\sim_{\mathcal{T}(U)}z$).

We use similar terminology for divisibility in the monoid $M$.

Definition 5.14**.**

Given an elements $a,c$ of $M$, we say that $c$ is divisible by $a$, and write $a|c$, if $c=ab$ for some $b\in M$ (perhaps not uniquely determined by $a$ and $c$). We say that $a$ and  $c$ are associated in $M$, and write $a\sim_{M}c$, if $a|c$ and $c|a.$ (If necessary, we write $a|_{M}c$ instead of $a|c$.) We also set $a\sim_{\mathcal{T}}c$ if $a\in\mathcal{T}(U)c$ and $c\in\mathcal{T}(U)a$.

It is obvious from Definition 5.11 that any son of a tyrant of $U$ is again a tyrant of $U$.

Theorem 5.15**.**

Assume that $x$ is a tyrant of $U$. Let $s_{x}:M\to U$ be the map defined by $s_{x}(c):=z$, if $x$ has a son $z$ with $ez=c,$ and $s_{x}(c):=c$ otherwise. Then $s_{x}$ is a primitive ig-section of $U$ with generator $x$.

Proof.

Proving that $s_{x}$ is an ig-section, it follows from Remark 5.12 that $s_{x}$ is primitive with generator $x$.

Obviously $es_{x}(c)=c$ for every $c\in M$. Given $a\in M$ and $y\in U$ with $s_{x}(a)y\in\mathcal{T}(U)$, it remains to verify that $s_{x}(a)y=s_{x}(ay)$, cf. Remark 5.2. Clearly $s_{x}(a)\in\mathcal{T}(U).$ Thus $z:=s_{x}(a)$ is a son of $x$, hence is again a tyrant. Since $zy\in\mathcal{T}(U),$ $zy$ is a son of $y$, hence is a son of $x$. We conclude that $zy=s_{x}(c)$ with $c:=ezy=ay$. On the other hand $zy=s_{x}(a)y$, and thus $s_{x}(a)y=s_{x}(ay).$ ∎

Theorem 5.16**.**

Assume that $s$ is an ig-section in $U$. Then every $x\in s(M)\cap\mathcal{T}(U)$ is a tyrant of $U$ and $s_{x}\leq s,$ $s_{x}(a)=s(a)=x$ for $s:=ex$. More generally, if $z$ is a son of $x$, then $s_{x}(ez)=s(ez).$

Proof.

Let $x\in s(M)\cap\mathcal{T}(U)$ and $a:=ex,$ hence $x=s(a).$ Assume that $z$ is a son of $x$ and $c:=ez$. Chose $y\in U$ with $z=xy.$ Then $c=exy=ay$ and $z=s(a)y=s(ay)=s(c),$ which proves that $x$ has exactly one son $z$ with $ez=c$, hence $x$ is a tyrant. Moreover $s(ez)=s_{x}(ez)$ for every son $z$ of $x$. In particular $(z=x)$, $s_{x}(a)=s(a).$ Given $c\in M$, if $x$ has no son $z$ with $ez=c,$ then $s_{x}(c)=c$. Thus $s_{x}(c)\in\{c,s(c)\}$ for every $c\in M$, proving that $s_{x}\leq s.$ ∎

Utilizing Proposition 5.8, we conclude the following from Theorem 5.16.

Corollary 5.17**.**

Let $s$ be a nontrivial ig-section in $U.$ Then $\{s_{x}\ {|}\ x\in s(M)\cap\mathcal{T}(U)\}$ is the set of all primitive ig-sections $t\leq s$. Its supremum in the poset $\operatorname{IGS}(U)$ is $s$.

{N.B. If $s$ is the trivial ig-section, this set is empty.}

Scholium 5.18**.**

Given an element $x$ of $\mathcal{T}(U)$, the following are equivalent.

1. (i)

$x$* is tyrant of $U$,* 2. (ii)

there exists an ig-section $s:M\to U$ with $x\in s(M),$ 3. (iii)

$x$* is a generator of a primitive ig-section in $U$.*

Proof.

$(iii)\Rightarrow(ii)$: trivial. $(ii)\Rightarrow(i)$: immediate by Theorem 5.16. $(i)\Rightarrow(iii)$: clear by Remark  5.12. ∎

We study the effect of a fiber contraction on ig-sections and tyrants.

Proposition 5.19**.**

Assume that $\alpha:U\twoheadrightarrow V$ is a fiber contraction over $M=eU=eV.$

1. (i)

If $s:M\to U$ is an ig-section on $U$, then $\alpha\circ s$ is an ig-section on $V$. 2. (ii)

Let $s:M\to U$ be a primitive ig-section with generator $x$. If $\alpha(x)$ is tangible, then $\alpha\circ s$ is primitive with generator $\alpha(x)$. Otherwise $\alpha\circ s$ is trivial. 3. (iii)

If $x$ is a tyrant of $U$ and $\alpha(x)$ is tangible, then $\alpha(x)$ is a tyrant of $V$ and $s_{\alpha(x)}=\alpha\circ s_{x}.$

Proof.

(i): We verify conditions $({\operatorname{S1}})$ and $({\operatorname{S2}})$ in Definition 5.1 for the section $s^{\prime}:=\alpha\circ s$. Clearly

[TABLE]

Furthermore, we have $s(M)U\subset M\cup s(M).$ Applying $\alpha$ we obtain $s^{\prime}(M)V\subset M\cup s^{\prime}(M).$ (ii): We work directly with Definition 5.10. Assume that $s$ is primitive with generator  $x$. Then $M\cup s(M)=M\cup xU$. Applying $\alpha$ we obtain

[TABLE]

(iii): Let $x$ be a tyrant of $U$ and $s:=s_{x}.$ As just proved $\alpha\circ s$ is primitive with generator  $\alpha(x).$ By Scholium 5.18 this means that $\alpha(x)$ is a tyrant of $V$ and $\alpha\circ s=s_{\alpha(x)}.$ ∎

It may happen that the supertropical monoid $U$ has no non-trivial ig-sections, then $U$ contains no tyrannic elements. However, in good cases, there exists a canonical way to produce tyrants by dividing out tangible MFCE-relations in $U$, to be explained in §7.

6. Equalizers

Let $U$ is a supertropical semiring with $M:=eU$, and let $S$ be an arbitrary subset of $U$. We look for MFCE-relations $E$ on $U$ such that $s\sim_{E}t$ for any two $s,t\in S$ with $es=et.$ The subset $\mathfrak{M}_{S}$ of $\operatorname{MFC}(U)$ consisting of these relations is certainly not empty, since it contains $E(\nu_{U}).$ Thus there exists a finest such relation $E$, namely the element $\bigwedge\mathfrak{M}_{S}$ in the complete lattice $\operatorname{MFC}(U).$

Definition 6.1**.**

If $S\subset U_{a}=\{x\in U\ {|}\ ex=a\}$ for some $a\in M$, we call $E$ the equalizer of the set $S$, and write $E=\operatorname{Eq}(S).$ In general we call $E$ the fiberwise equalizer of $S$, and write $E=\operatorname{Feq}(S).$

Recall that for any $X\subset U$ and $c\in M$ we write

[TABLE]

Remark 6.2**.**

It is obvious from the definition that

[TABLE]

Here and in all the following we do not exclude the case that $S$ (or some $S_{c}$) is empty. If $S=\emptyset$ then by definition $\operatorname{Feq}(S)=\operatorname{diag}(U).$ This also holds if $S$ is a one-point set.

We strive for an explicit description of $\operatorname{Feq}(S)$. First an easy case.

Proposition 6.3**.**

Assume that $a\in S\subset U_{a}.$ Then $\operatorname{Eq}(S)$ is the relation $E(U,S):=E(U,M\cup US)$ which compresses the ideal $M\cup US$ to ghosts (cf. [IKR4, Corollary 1.17]).

Proof.

Making the elements of $S$ equal means identifying them with their common ghost companion $a$, then every element of $US$ is identified with its ghost companion. ∎

Definition 6.4**.**

An $S$-path $\gamma$ of length in $U$ is a finite sequence $z_{1},\dots,z_{n+1}$ of elements of $U$, called the nodes of $\gamma$, together with a sequence of triples

[TABLE]

called the labels of $\gamma$, such that $es_{k}=et_{k}$ for $1\leq k\leq n$ and

[TABLE]

We say that $\gamma$ is an elementary $S$-path, if $n=1$. Given $z,w\in U$, $\gamma$ an $S$-path connecting $z$ to $w$ (or: from $z$ to $w$) if $z=z_{1},w=z_{n+1}$.

Notice that, given an $S$-path $\gamma$, every “inner” node $z_{k}$ ($2\leq k\leq n$) is presented as a product of an element of $U$ and an element of $S$ in two different ways,

[TABLE]

Notice also that $ez_{1}=\cdots=ez_{n+1}.$ Observe finally that our notation of an $S$-path  $\gamma$ contains a redundancy: The sequence of labels determines the sequence of nodes.

In the following it may be help to visualize $S$-paths by diagrams, where multiplication by an element $u\in\mathcal{T}(U)$ is presented by an arrow $\overset{u}{\longrightarrow}$. For example, the diagram of an $S$-path $\gamma$ with $3$ nodes ($n=2$).

[TABLE]

Definition 6.5**.**

Let $\gamma$ be an $S$-path from $z$ to $w$ with sequence of labels (6.1). We obtain the inverse $S$-path $\gamma^{-1}$ from $w$ to $z$ by changing the sequence of labels (6.1) to

[TABLE]

Furthermore, if $\gamma^{\prime}$ is an $S$-path from $w$ to $w^{\prime}$ we obtain a path $\gamma*\gamma^{\prime}$ from $z$ to $w^{\prime}$ by juxtaposing the sequence of labels of $\gamma^{\prime}$ to the sequence of labels of $\gamma$.

Notations 6.6**.**

Henceforth we denote an elementary $S$-path simply by its labels, and an $S$-path $\gamma$ with sequence of labels (6.1) as

[TABLE]

In this notation we have

[TABLE]

For later use we quote the following obvious fact.

Remark 6.7**.**

Given an $S$-path $\gamma$ with sequence of labels (6.1) and an element $u^{\prime}$ of $U$, we obtain a new $S$-path

[TABLE]

If $\gamma$ connects $z$ to $w$ then $u^{\prime}\gamma$ connects $u^{\prime}z$ to $u^{\prime}w$.

Given an element $z_{0}$ of the monoid ideal

[TABLE]

generated by $S$, and choosing $u_{0}\in U$, $s_{0}\in S$ with $z_{0}=u_{0}s_{0}$, we get the elementary path $(s_{0},u_{0},s_{0})$. When $z_{0}\notin US,$ there is no $S$-path which starts or ends at  $z_{0}.$

Definition 6.8**.**

Two elements $z,w$ of $US$ are called $S$-connected, written $z\sim_{S}w,$ if there exists an $S$-path from $z$ to $w.$

By the above discussion it is evident that “$S$-connected” is an equivalence relation on the set $US$, whose equivalence classes are called the $S$-components of $US$ (or: of $U$).

Theorem 6.9**.**

The equivalence classes of the MFCE-relation $\operatorname{Feq}(S)$ are the $S$-components of $US$ and the one-element sets $\{z\}$ with $z\in U\setminus\{US\}.$

Proof.

We define an equivalence relation $E$ on $U$ by $z\sim_{E}w$ iff either $z=w$ or there exists an $S$-path from $z$ to $w$. We learn from Remark 6.7 that $z\sim_{E}w$ implies $uz\sim_{E}uw$ for any $u\in U.$ As observed above, every $S$-path runs in a fiber $U_{c},$ $c\in M$. Thus $z\sim_{E}w$ implies also $ez=ew$. This proves that $E$ is an MFCE-relation. If $s,t\in S,$ the elementary $S$-path $(s,1,t)$ runs form $s$ to $t$, hence $s\sim_{E}t.$

Assume that $F$ is any MFCE-relation on $U$ with $s\sim_{F}t$ for all $s,t\in S.$ Given an elementary $S$-path $(s,u,t)$ we conclude form $s\sim_{F}t$ that $us\sim_{F}ut.$ It follows that any two $S$-connectable elements of $US$ are $F$-equivalent, hence $F$ is coarser than $E$, proving that $E=\operatorname{Feq}(S).$ ∎

These arguments give us a constructive proof for the existence of the fiberwise equalizer $\operatorname{Feq}(S)$, which does not use the fact that $\operatorname{MFC}(U)$ is a complete lattice.

Corollary 6.10**.**

Let $S$ be any subset of $U$. The following are equivalent.

1. (i)

$\operatorname{Feq}(S)$* is ghost separating.* 2. (ii)

The nodes of each $S$-path are in one of the sets $\mathcal{T}(U)$, $\mathcal{G}(U)$, $\{0\}.$ 3. (iii)

For every elementary $S$-path $(s,u,t)$ both $us$ and $ut$ are contained in one of the sets $\mathcal{T}(U)$, $\mathcal{G}(U)$, $\{0\}.$ 4. (iv)

For every $u\in U$ and $c\in M$ either $uS_{c}\subset\mathcal{T}(U)$, or $uS_{c}\subset\mathcal{G}(U)$, or $uS_{c}=\{0\}.$

Proof.

(i) $\Leftrightarrow$ (ii): This is clear from Theorem 6.9. (ii) $\Leftrightarrow$ (iii): Obvious. (iii) $\Rightarrow$ (iv): Given $u\in U$, assume that $S_{c}\neq\emptyset$, and fix some $s_{0}\in S_{c}.$ Apply (iii) to the elementary path $(s_{0},u,s)$ with $s$ running through $S_{c}.$ The element $us_{0}$ is contained in one of the sets $\mathcal{T}(U)$, $\mathcal{G}(U)$, $\{0\}$. It follows from (iii) that $uS_{c}$ is contained in the same set. (iv) $\Rightarrow$ (iii): Fix an elementary $S$-path $(s,u,t),$ and let $c:=es=et.$ By (iv), the set $uS_{c}$ is contained in one of the sets $\mathcal{T}(U)$, $\mathcal{G}(U)$, $\{0\},$ implying that $\{us,ut\}$ is a subset of the same set. ∎

In a similar way we handle MFCE-relations which equalize fiberwise all members of a family of subsets of $U$ instead of a single subset of $U$. The following is obvious.

Proposition 6.11**.**

Let $\mathfrak{S}:=(S_{\lambda}\ {|}\ \lambda\in\Lambda)$ be a family of subsets of $U$. Then

[TABLE]

is the finest MFCE-relations $E$ on $U$ such that, if $s,t\in S_{\lambda}$ for some $\lambda\in\Lambda$ and $es=et$, then $s\sim_{E}t.$

This relation $\operatorname{Feq}(\mathfrak{S})$ can be described by “paths” in a way analogous to Theorem 6.9.

Definition 6.12**.**

An elementary $\mathfrak{S}$-path is a triple $(s,u,t)\in U\times U\times U$ with $es=et$ and $\{s,t\}\in S_{\lambda}$ for some $\lambda\in\Lambda.$ Consequentially an $\mathfrak{S}$-path $\gamma$ is a sequence of such triples $(s_{k},u_{k},t_{k})$, $1\leq k\leq n$, written

[TABLE]

such that the equations (6.2) from above are valid. Again we call the elements

[TABLE]

the nodes of $\gamma$, and say that $\gamma$ runs from $z_{1}$ to $z_{n+1}.$

Notice that all nodes $z_{k}$ are elements of the monoid ideal $\bigcup_{\lambda\in\Lambda}US_{\lambda}.$ In particular, if $z_{0}\in U$ is not in this set, there exists no $\mathfrak{S}$-path starting or ending at $z_{0}.$

Theorem 6.13**.**

If $z,w\in U$ and $z\neq w$, then $z$ and $w$ are $\operatorname{Feq}(\mathfrak{S})$-equivalent iff there exists an $\mathfrak{S}$-path form $z$ to $w.$

Proof.

Since $\operatorname{Feq}(\mathfrak{S})=\bigvee_{\lambda\in\Lambda}\operatorname{Feq}(S_{\lambda})$ is the equivalence relation on $U$ generated by the relations $\operatorname{Feq}(S_{\lambda})$ (cf. [IKR1, §7]), the given elements $z\neq w$ are $\operatorname{Feq}(\mathfrak{S})$-equivalent iff there exists a sequence $z=x_{1},x_{2},\dots,x_{N}=w$ in $U$ and indices $\lambda(1),\dots,\lambda(N-1)$ in $\Lambda$ such that $x_{i}\neq x_{i+1}$ for $1\leq i\leq N-1$ and

[TABLE]

If this holds, then by Theorem 6.9 there exists an $S_{\lambda(i)}$-path $\gamma_{i}$ form $x_{i}$ to $x_{i+1}$, and $\gamma:=\gamma_{1}*\cdots*\gamma_{N}$ is an $\mathfrak{S}$-path from $z$ to $w.$

Conversely, given an $\mathfrak{S}$-path $\gamma$ from $z$ to $w$ with nodes $z_{1},\dots,z_{n+1}$, write $\gamma:=\gamma_{1}*\cdots*\gamma_{n}$, where each $\gamma_{k}$ is elementary from $z_{k}$ to $z_{k+1}.$ Omitting the $\gamma_{k}$ with $z_{k}=z_{k+1}$, we are in the situation $(*)$, with $N=n+1$, $x_{i}=z_{i}.$ ∎

In complete analogy to the proof of Corollary 6.10 we obtain

Corollary 6.14**.**

Let $\mathfrak{S}$ be a family of subsets of $U$. The following are equivalent:

1. (i)

$\operatorname{Feq}(\mathfrak{S})$* is ghost separating.* 2. (ii)

The nodes of each $\mathfrak{S}$-path are in one of the sets $\mathcal{T}(U)$, $\mathcal{G}(U),$ $\{0\}$. 3. (iii)

For every $u\in U,$ $c\in M,$ $\lambda\in\Lambda$ the set $u(S_{\lambda})_{c}$ is contained in one of the sets $\mathcal{T}(U)$, $\mathcal{G}(U),$ $\{0\}$.

7. Producing isolated tangibles and tyrants

As before $U$ is a supertropical semiring and $M:=eU$.

Notations 7.1**.**

**

1. (a)

Given $x,z\in U,$ we say that $x$ divides $z$ (in $U$), written $x|z$, if $z\in Ux.$ Otherwise we write $x\nmid z$. (N.B. If $a,c\in M$, then divisibly of $c$ by $a$ means the same in $M$ as in  $U$, since $c\in Ua$ iff $c\in Ma$.) 2. (b)

We define

[TABLE]

Of course, if $x\nmid z$ then $[z:x]=\emptyset.$ 3. (c)

Given $x\in\mathcal{T}(U),$ we define

[TABLE]

and call, as already done in §2, the elements of $\SS(x)$ the sons of $x$. The sons $z\neq x$ are said to be proper sons of $x$. We let

[TABLE]

and call the elements of $\SS_{c}(x)$ the sons of $x$ over $c$. 4. (d)

We denote the fiberwise equalizer $\operatorname{Feq}(\SS(x))$ (cf. §3) by $\operatorname{T}(x)$ and the equalizer $\operatorname{Eq}(\SS_{c}(x))$ by $\operatorname{T}_{c}(x).$

We explicate the meaning of these MFCE-relations, first of $\operatorname{T}_{c}(x)$ and then of $\operatorname{T}(x).$ Studying $\operatorname{T}_{c}(x)$ we may assume that $c\in\mathcal{T}(U)a,$ since otherwise $\SS_{c}(x)$ is empty. The following remains true without this assumption.

Proposition 7.2**.**

Let $x\in\mathcal{T}(U)$ and $c\in M.$

1. (a)

$\operatorname{T}_{c}(x)$* is the finest MFCE-relation $E$ on $U$ such that $[x]_{E}$ is ghost or $[x]_{E}$ is tangible and has at most one son over $c$ in $U/E$. (Recall that we identify $M/E=M.$)* 2. (b)

More explicitly we have the following alternative.

Case I:

$\operatorname{T}_{c}(x)$* is the finest MFCE-relation $E$ on $U$ such that $[x]_{E}$ is tangible and has exactly one son over $c$ in $U/E.$*

Case II:

There exists no such relation $E$. Then $\operatorname{T}_{c}(x)=E(U,\SS_{c}(x)).$

Proof.

These claims are essentially trivial. First observe that for any MFCE-relation $E$ on  $U$ we have

[TABLE]

Thus $\operatorname{T}_{c}(x)=\operatorname{Eq}(\SS_{c}(x))$ is the finest MFCE-relation $E$ such that either $[x]_{E}$ is ghost, then $[z]_{E}=c$ for all $z\in\SS_{c}(x)$), or $[x]_{E}$ is tangible and has at most one son over $c$. Case  II happens iff $\operatorname{T}_{c}(x)$ identifies all elements of $\SS_{c}(x)$ with the ghost $c$. Proposition 6.3 gives $\operatorname{T}_{c}(x)=E(U,\SS_{c}(x)).$ ∎

By analogous arguments we obtain

Proposition 7.3**.**

Let $x\in\mathcal{T}(U)$.

1. (a)

$\operatorname{T}(x)$* is the finest MFCE-relation $E$ on $U$, such that $[x]_{E}$ is a ghost, or $[x]_{E}$ is a tyrant in $U/E$ (i.e., $[x]_{E}$ is tangible, and for any $c\in M$ has at most one son over  $c$, cf.  §2).* 2. (b)

More explicitly we have the following cases.

Case I:

$\operatorname{T}(x)$* is the finest MFCE-relation $E$ on $U$ such that $[x]_{E}$ is a tyrant in $U/E.$*

Case II:

There exists no such relation. Then $\operatorname{T}(x)=E(U,\SS(x)).$

Theorem 7.4**.**

Let $x\in\mathcal{T}(U),$ $a:=ex.$ The following are equivalent.

1. (i)

$[x]_{\operatorname{T}(x)}$* is ghost in $U/\operatorname{T}(x).$* 2. (ii)

There exists an elementary $\SS(x)$-path $(vx,u,wx)$ from a son $z=uvx$ of $x$ over $a$ to $ez=a.$ 3. (iii)

There exist elements $u,v,w$ of $U$ with $evx=ewx,$ $euvx=ex,$ $wx\in\mathcal{T}(U),$ $uvx\in\mathcal{T}(U),$ $uwx\in M.$

Proof.

(i) $\Rightarrow$ (ii): (i) means that there exists an $\SS(x)$-path

[TABLE]

from $x$ to $ex$ (cf. Theorem 6.9). We choose $\gamma$ of minimal length. Then $u_{n}v_{n}x\in\mathcal{T}(U)$, $u_{n}w_{n}x\in M.$ We have $u_{1}v_{1}x=x,$ hence

[TABLE]

for each $i\in\{1,\dots,n\}.$ Thus $z:=u_{n}v_{n}x$ is a son of $x$ over $a$, and $(v_{n}x,u_{n},w_{n}x)$ is an elementary $\SS(x)$-path from $z$ to $ez=a$. (ii) $\Rightarrow$ (i): $\gamma:=(x,1,z)*(vx,u,wx)$ is an $\SS(x)$-path form $x$ to $a.$ Thus $[x]_{\operatorname{T}(x)}=[ex]_{\operatorname{T}(x)}$ $(=a)$ is ghost. (ii) $\Leftrightarrow$ (iii): Condition (iii) means that $(vx,u,wx)$ is an $\SS(x)$-path from $z=uvx$ to $ez$, and $z$ is a son of $x$ over $a.$ ∎

Scholium 7.5**.**

We can reformulate the equivalence (i) $\Leftrightarrow$ (ii) in Theorem 7.4 as: Given $x\in\mathcal{T}(U),$ the element $[x]_{\operatorname{T}(x)}$ is tangible, hence a tyrant, iff for all $u,v,w\in U$ with $evx=ewx$, $euvx=ex,$ and both $wx$ and $uvx$ tangible, also the product $uwx$ is tangible.

Exploiting the contents of Corollary 6.10 for $\SS=\SS(x)$, we obtain a criterion that $[z]_{\operatorname{T}(x)}$ is tangible for every son $z$ of $x$.

Theorem 7.6**.**

Let $x\in\mathcal{T}(U).$ The following are equivalent.

1. (i)

The MFCE-relation $\operatorname{T}(x)$ is tangible. 2. (ii)

For every son $z$ of $x$ in $U$ the element $[z]_{\operatorname{T}(x)}$ is tangible in $U/\operatorname{T}(x)$ (and hence a tyrant). 3. (iii)

If $u,v,w\in U$ and $evx=ewx,$ $uvx\in\mathcal{T}(U),$ $wx\in\mathcal{T}(U),$ then $uwx\in\mathcal{T}(U).$

Proof.

(i) $\Leftrightarrow$ (ii): This is a consequence of a general fact for TE-relations. If $E$ is a TE-relation on $U$ and $z\in U$ then $[z]_{E}$ is ghost iff $[z]_{E}=e[z]_{E}=[ez]_{E}$, i.e., $z\sim_{E}ez$. Thus $[z]_{E}$ is tangible for every $z\in\mathcal{T}(U)$ iff $z\nsim_{E}ez$ for every $z\in\mathcal{T}(U),$ which means that $E$ is tangible (= ghost separating) when $E$ is MFCE. In the present case, where $E=\operatorname{T}(x),$ we may focus on the elements $z\in\SS(x),$ since for $z\notin\SS(x)$ we have $[z]_{E}=\{z\},$ hence $z\nsim_{E}ez$ for $z$ tangible.

(ii) $\Leftrightarrow$ (iii): By (i) $\Leftrightarrow$ (iii) of Corollary 6.10 we know that $\operatorname{T}(x)$ is ghost separating iff, for any $u,v,w\in U$ with $vx,wx\in\mathcal{T}(U),$ $evx=ewx,$ the set $\{uvx,uwx\}$ is contained in $\mathcal{T}(U)$ or in $M$. Now, if $ux\in M,$ then trivially $\{uvx,uwx\}\subset M.$ Thus we may focus on elements $u$ with $ux\in\mathcal{T}(U)$, and see that condition (iii) in Corollary 6.10 translates to condition (iii) in the present claim. ∎

Example 7.7**.**

Let $x\in\mathcal{T}(U).$ Assume that the product of any two sons of $x$ in $U$ is tangible. Then, for every son $z$ of $x$, $[z]_{\operatorname{T}(x)}$ is a tyrant in $U/\operatorname{T}(x)$. Indeed, we conclude from $uvx\in\mathcal{T}(U)$ and $wx\in\mathcal{T}(U),$ that $uvx\cdot wx\in\mathcal{T}(U),$ hence $uwx\in\mathcal{T}(U).$

We turn to a property of tangible elements, which is weaker than being a tyrant, but seems to have equal importance.

Definition 7.8**.**

Let $x\in\mathcal{T}(U)$ and $a:=ex.$

1. (a)

We call $x$ isolated (in $U$) if $\SS_{a}(x)=\{x\}$, i.e., $x$ has no proper son over $a$. 2. (b)

We call $\operatorname{T}_{a}(x)=\operatorname{Eq}(\SS_{a}(x))$ the isolating MFCE-relation (on $U$) for $x$ and denote it by $\operatorname{Is}(x).$

Proposition 7.2 tells us that $\operatorname{Is}(x)$ is the finest MFCE-relation $E$ on $U$, such that either $[x]_{E}\in\mathcal{T}(U/E)$ and $[x]_{E}$ is isolated (Case I), or $[x]_{E}$ is ghost (Case II). When the former case holds, we say that $x$** can be isolated** (in $U$), otherwise we say that $x$ cannot be isolated. In the latter case $\operatorname{Is}(x)=E(U,x)$.

Theorem 7.9**.**

Let $x\in\mathcal{T}(U)$ and $a:=ex.$ The following are equivalent.

1. (i)

$x$* cannot be isolated in $U$.* 2. (ii)

There exist $u,v,w\in[a:a]_{U}$ such that $uvx\in\mathcal{T}(U),$ $wx\in\mathcal{T}(U),$ but $uwx\in M$ (hence $uwx=a$).

Proof.

(ii) $\Rightarrow$ (i): Let $E:=\operatorname{Is}(x)=\operatorname{Eq}(\SS_{a}(x))$. Since $uvx,vx,wx\in\SS_{a}(x)$, we conclude  that

[TABLE]

(i) $\Rightarrow$ (ii): There exists an $\SS_{a}(x)$-path

[TABLE]

of shortest length $n$ connecting $x$ to $a$. Then $u_{n}v_{n}x\in\mathcal{T}(U)$ and $u_{n}w_{n}x=a.$ The elements $v_{i}x$, $w_{i}x$ lie in $\SS_{a}(x)$ and $x=u_{1}v_{1}x\in\SS_{a}(x).$ It follows that $a=v_{i}a=w_{i}a,$ i.e., $v_{i},w_{i}\in[a:a].$ Furthermore

[TABLE]

and thus all $u_{i}\in[a:a].$ Condition (ii) holds with $u=u_{n},$ $v=v_{n}$, $w=w_{n}.$ ∎

Example 7.10**.**

Let $a\in M\setminus\{0\}.$ Assume that the set $[a:a]_{U}\cap\mathcal{T}(U)$ is closed under multiplication. Then condition (ii) in Theorem 7.9 is violated for every $x\in\mathcal{T}(U)_{a}.$ Thus every $x\in\mathcal{T}(U)_{a}$ can be isolated.

It turns out that the isolating relation $\operatorname{Is}(x)$ does not alter if we replace $x$ by an associated element (cf. Definition 5.13).

Lemma 7.11**.**

Assume that $x,x^{\prime}\in\mathcal{T}(U)$ are associated, i.e., $x|x^{\prime}$ and $x^{\prime}|x.$ Then, for any  $u\in U$,

1. (i)

$ux\in\mathcal{T}(U)\Leftrightarrow ux^{\prime}\in\mathcal{T}(U),$** 2. (ii)

$eux=ex\Leftrightarrow eux^{\prime}=ex^{\prime}.$**

Proof.

In both claims it suffices to verify the direction $\Rightarrow.$ Given elements $v,v^{\prime}$ of $U$ with $x^{\prime}=vx,$ $x=v^{\prime}x^{\prime}$, if $ux\in\mathcal{T}(U),$ then $uv^{\prime}x^{\prime}\in\mathcal{T}(U),$ hence $ux^{\prime}\in\mathcal{T}(U).$ If $eux=ex,$ then

[TABLE]

∎

Proposition 7.12**.**

Assume that $x^{\prime}\in\mathcal{T}(U)$ is associated to $x\in\mathcal{T}(U).$ Then

1. (i)

$\operatorname{Is}(x)=\operatorname{Is}(x^{\prime}),$** 2. (ii)

In particular, $x$ is isolated iff $x^{\prime}$ is isolated, 3. (iii)

$x$* can be isolated iff $x^{\prime}$ can be isolated.*

Proof.

Choose again $v,v^{\prime}\in U$ with $x^{\prime}=vx,$ $x=v^{\prime}x^{\prime}$, and let $a:=ex,$ $a^{\prime}:=ex^{\prime}.$

(i): We learn from Lemma 7.11 that $\SS_{a^{\prime}}(x^{\prime})=v\SS_{a}(x^{\prime}).$ Thus $\operatorname{Eq}(\SS_{a}(x))\supset\operatorname{Eq}(\SS_{a^{\prime}}(x^{\prime})).$ By symmetry these equivalence relations are equal, i.e., $\operatorname{Is}(x)=\operatorname{Is}(x^{\prime}).$ (ii): In this situation $\operatorname{Is}(x)=\operatorname{Is}(x^{\prime})=\operatorname{diag}(U).$ (iii): Let $E:=\operatorname{Is}(x)=\operatorname{Is}(x^{\prime}).$ From $x|x^{\prime}$ and $x^{\prime}|x$ we conclude that $[x]_{E}\ {|}\ [x^{\prime}]_{E}$ and $[x^{\prime}]_{E}\ {|}\ [x]_{E}$. Thus $[x]_{E}$ is tangible iff $[x^{\prime}]_{E}$ is tangible. ∎

We are interested in cases where a tangible element and all its sons are simultaneously isolated.

Definition 7.13**.**

Let $x\in\mathcal{T}(U)$. We write

[TABLE]

and call it the son isolating MFCE-relation for $x$ (on $U$).

In other terms, $\operatorname{Sis}$ is the fiberwise equalizer $\operatorname{Feq}(\mathfrak{S}(x))$ of the family

[TABLE]

of subsets of $U.$ This is the finest MFCE-relation $E$ on $U$ such that for every son $z$ of $x$ the element $[z]_{E}$ is either ghost or isolated tangible in $U/E$.

Remark 7.14**.**

We have the following cases.

Case I:

$[x]_{\operatorname{Sis}(x)}$* is tangible. Then all sons of $[x]_{\operatorname{Sis}(x)}$ in $U/{\operatorname{Sis}(x)}$ are isolated.*

Case II:

$[x]_{\operatorname{Sis}(x)}$* is ghost. Then ${\operatorname{Sis}(x)}=E(U,x).$*

Notations 7.15**.**

**

1. (a)

If $z\in\mathcal{T}(U),$ $c:=ez$, we call an $\SS_{c}(z)$-path also an $\operatorname{Is}(z)$-path. Thus, if $z$ is a son of $x\in\mathcal{T}(U)$, $z=px$, an elementary $\operatorname{Is}(z)$-path is a triple $(vpx,u,wpx)$ with $u,v,w,p\in U,$ $vpx\in\mathcal{T}(U),$ $wpx\in\mathcal{T}(U),$ and $evpx=ewpx=epx.$ 2. (b)

If $x\in\mathcal{T}(U)$, we call an $\mathfrak{S}(x)$-path (cf. Definition 6.12) also an $\operatorname{Sis}(x)$-path. Thus, an elementary $\operatorname{Sis}(x)$-path is an elementary $\operatorname{Is}(z)$-path for some son $z=px$ of $x$.

As in the proof of Theorem 6.13, but replacing $\mathfrak{S}$-paths by $\mathfrak{S}(x)$-paths, we obtain

Theorem 7.16**.**

Let $x\in\mathcal{T}(U)$ and $a:=ex.$ The following are equivalent.

1. (i)

$[x]_{\operatorname{Sis}(x)}$* is ghost in $U/{\operatorname{Sis}(x)}$.* 2. (ii)

There exists a son $z$ of $x$ and an elementary $\operatorname{Is}(z)$-path $(vz,u,wz)$ from a son $uvz$ of $z$ over $a$ to $a$. (N.B. $vz,wz\in\mathcal{T}(U)$.) 3. (iii)

There exists $u,v,w,p\in U$ with $epx=evpx=ewpx,$ $euvpx=a$, $uvpx\in\mathcal{T}(U)$, $wpx\in\mathcal{T}(U)$, $uwpx=a.$

Arguing again as in the proof of Theorem 6.13, we obtain

Theorem 7.17**.**

Let $x\in\mathcal{T}(U)$. The following are equivalent.

1. (i)

The MFCE-relation ${\operatorname{Sis}(x)}$ is tangible. 2. (ii)

For every son $z$ of $x$ in $U$ the element $[z]_{\operatorname{Sis}(x)}$ is tangible in $U/\operatorname{Sis}(x)$ (and hence isolated). 3. (iii)

If $u,v,w,p$ are elements of $U$ with $evpx=ewpx=epx,$ $uvpx\in\mathcal{T}(U)$, $wpx\in\mathcal{T}(U)$, then $uwpx\in\mathcal{T}(U).$

Of course $\operatorname{Is}(x)\subset\operatorname{Sis}(x)$. We now exhibit a good case, where these two MFCE-relations coincide. This means that for every son $z$ of $x$ the element $[z]_{\operatorname{Is}(x)}$ is either ghost or tangible in $U/\operatorname{Is}(x)$.

Theorem 7.18**.**

Let $x\in\mathcal{T}(U)$ and $a:=ex.$ Assume that the submonoid $\mathcal{T}(U)e$ of $M$ admits the cancellation hypothesis: for any $b,c,d\in\mathcal{T}(U)e$

[TABLE]

(N.B. This certainly holds if $\mathcal{T}(U)e$ is cancellative.) Then

[TABLE]

Proof.

Let $\gamma=(vpx,u,wpx)$ be an elementary $\operatorname{Sis}(x)$-path with $u,v,w,p\in U$, $vpx,wpx\in\mathcal{T}(U)$, $evpx=ewpx=epx.$ Then, by the cancellation hypothesis, $evx=ewx=ex$. Thus $\widetilde{\gamma}:=(vx,up,wx)$ is an $\operatorname{Is}(x)$-path with the same pair of nodes as $\gamma.$ It follows that, given an $\operatorname{Sis}(x)$-path form $z\in U$ to $w\in U,$ there also exists an $\operatorname{Is}(x)$-path from $z$ to $w.$ Thus $\operatorname{Sis}(x)\subset\operatorname{Is}(x)$, which trivially implies $\operatorname{Sis}(x)=\operatorname{Is}(x)$. ∎