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Victoria Gould

Publications and source records attributed to Victoria Gould.

At least 19 recordsLinked to original sources
arXiv2026

The endomorphism tower of a finite symmetric group

We consider the endomorphism tower of a monoid $M$, that is, the sequence of monoids End$_i(M)$ where End$_0(M)=M$ and for all $i\geq 1$, End$_i(M)$ is the monoid of all endomorphisms of End$_{i-1}(M)$. We show that for a finite monoid $M$ this sequence does not stabilise in a finite number of steps. Our focus is then on the case where $M=\mathcal{S}_n$, the symmetric group on a finite number $n$ of points. It is well known that other than in exceptional cases (which are avoided by taking $n \geq 7$), the corresponding automorphism tower of $\mathcal{S}_n$ stabilises at the first step. In spite of the natural nature of this question, nothing was known of the endomorphism tower above the level $i=1$. We determine (for each $n \geq 7)$ the elements of End$_2(\mathcal{S}_n)$ and their multiplication and thus verify that the monoids End$_i(\mathcal{S}_n)$ for $i=0,1,2$ all have group of units isomorphic to $\mathcal{S}_n$. We show that the same is true of End$_3(\mathcal{S}_n)$.

math.GR↗
arXiv2026

Left Ehresmann monoids with a proper basis

Left Ehresmann monoids, and their two-sided counterpart of Ehresmann monoids, were so named by Lawson, who elucidated their connection to the work of Ehresmann in differential geometry. This article is dedicated to building a theory for left Ehresmann monoids inspired by that for inverse semigroups; in order to do so we must develop substantially different ideas and techniques. It is known that every left Ehresmann monoid has a cover, that is, a projection separating preimage, of the form $\mathcal{P}_{\ell}(T,X)$, where $\mathcal{P}_{\ell}(T,X)$ is a left Ehresmann monoid constructed from a monoid $T$ and an order-preserving action of $T$ on a semilattice $X$ with identity. We introduce the notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$, and consequently any free left Ehresmann monoid, possesses a proper basis. We show that any left Ehresmann monoid with a proper basis displays properties close to those of two-sided Ehresmann monoids. Next, we exhibit a class of subsemigroups $\mathcal{Q}_{\ell}(T,X,Y)$ (properly, biunary monoid subsemigroups) of the monoids $\mathcal{P}_{\ell}(T,X)$, which are also left Ehresmann with a proper basis. We prove that any left Ehresmann monoid with a proper basis is isomorphic to some $\mathcal{Q}_{\ell}(T,X,Y)$. Our results can be regarded as being analogous to those for proper inverse semigroups, due to McAlister and O'Carroll, the $\mathcal{Q}_{\ell}(T,X,Y)$ playing the role of the $P$-semigroups and the $\mathcal{P}_{\ell}(T,X)$ the role of the semidirect products of a semilattice by a group. In the process of proving our main theorems we present a globalisation result for an order-preserving partial action of a monoid on a partially ordered set or semilattice.

math.RA↗
arXiv2025

Subset expansions of monoids

We initiate the study of the expansion $\mathcal{S}(M)$ of a monoid $M$ obtained via the semidirect product of $M$ acting naturally on the left of its power set (regarded as a semilattice under union). We term this the `subset expansion' of $M$. The monoid $\mathcal{S}(M)$ contains the images of several expansions of $M$ of wide interest and use in semigroup theory, in particular the prefix and Szendrei expansions (in the case where $M$ is free, these `smaller' expansions produce free algebras in certain varieties). We first focus on algebraic properties, specifically those determined by idempotents. Particularly, we show that the expansion $\mathcal{S}$ maps groups to proper inverse monoids, unipotent monoids to proper left restriction monoids, right cancellative monoids to left ample monoids, right abundant monoids to right abundant monoids, and left cancellative monoids to right adequate monoids. Subsequently, we focus on finitary conditions. We examine the condition of weak left coherence (every finitely generated left ideal has a finite presentation as a left act); the related conditions of property (L), left ideal Howson, finitely left equated, and each of the corresponding left-right dual notions. Each of these conditions is preserved under retract, from which it is immediate that if $\mathcal{S}(M)$ satisfies one of our finitary conditions, then so must $M$, but the converse is not true. For a property to `lift' from $M$ to $\mathcal{S}(M)$ it must undergo a strengthening. Indeed, we show that $\mathcal{S}(M)$ satisfies property (L) (or its left-right dual) if and only if $M$ is finite. We provide exact characterisations of the monoids $M$ such that $\mathcal{S}(M)$ is: left (or right) ideal Howson; finitely left equated; and (consequently) weakly left coherent. We give sufficient conditions for $\mathcal{S}(M)$ to be finitely right equated and hence weakly right coherent.

math.RA↗
arXiv2025

Finitary conditions for graph products of monoids

Graph products of monoids provide a common framework for free products and direct products. Trace monoids are graph products of finitely generated free monoids. We investigate the interaction of certain finitary conditions with graph products. Specifically, we examine the conditions of being weakly left noetherian (that is, every left ideal is finitely generated) and weakly left coherent (that is, every finitely generated left ideal has a finite presentation) and the related conditions of the ascending chain condition on principal left ideals, being left ideal Howson, and being finitely left equated. All of these conditions, and others, are preserved under retract; as a consequence, if a graph product has such a property, then so do all the constituent monoids. We show that the converse is also true for all the conditions listed except that of being weakly left noetherian. In the latter case we precisely determine the graph products of monoids which are weakly left noetherian.

math.RA↗
arXiv2025

Forbidden configurations for coherency

Right (and left) coherency and right (and left) weak coherency are natural finitary conditions for monoids. Determining whether or not a given monoid has any of these properties is historically a difficult problem. This paper has several aims, centering around the well-studied class of right (and dually left) $E$-Ehresmann monoids, being one of the broadest classes of monoids containing a semilattice of idempotents. First, we exhibit a particular configuration of elements in a monoid subsemigroup of a right (respectively, left) $E$-Ehresmann monoid, relative to the Ehresmann structure of the overmonoid, that prohibits left (respectively, right) coherence. Second, we apply this technique in a number of different situations. We show that the free Ehresmann monoid of rank at least $2$ is neither left nor right coherent, and that the free left Ehresmann monoid is not left coherent. We demonstrate the utility of our technique in the case where the overmonoid is an $E$-unitary inverse monoid, and apply this to both new situations and to recover the existing results. Namely, free inverse monoids and free ample monoids of rank at least 2 are neither left nor right coherent, and free left ample monoids of rank at least $2$ is are not left coherent. Next, in a positive direction, we demonstrate that every free left Ehresmann monoid is weakly coherent. Our final result is of a different nature. Ehresmann monoids form a variety of monoids with an enriched signature. Viewed as a bi-unary monoid (respectively, unary monoid), a free Ehresmann monoid (respectively, free left Ehresmann monoid) does not embed into an inverse monoid. We show that viewed as a monoid (the standpoint of this paper) every free Ehresmann monoid (and hence also every free left Ehresmann monoid) embeds into an $E$-unitary inverse monoid.

math.RA↗
arXiv2024

Weakly right coherent monoids

A monoid $S$ is said to be weakly right coherent if every finitely generated right ideal of $S$ is finitely presented as a right $S$-act. It is known that $S$ is weakly right coherent if and only if it satisfies the following conditions: $S$ is right ideal Howson, meaning that the intersection of any two finitely generated right ideals of $S$ is finitely generated; and the right annihilator congruences of $S$ are finitely generated as right congruences. We examine the behaviour of these two conditions (in the more general setting of semigroups) under certain algebraic constructions and deduce closure results for the class of weakly right coherent monoids. We also show that the property of being right ideal Howson is related to the axiomatisability of a class of left acts satisfying a condition related to flatness.

math.RA↗
arXiv2024

Diameters of endomorphism monoids of chains

The left and right diameters of a monoid are topological invariants defined in terms of suprema of lengths of derivation sequences with respect to finite generating sets for the universal left or right congruences. We compute these parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically, if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a quotient of $C{\setminus}\{z\}$ for some endpoint $z$. If $C$ is finite then so is $End(C),$ in which case the left and right diameters are 1 (if $C$ is non-trivial) or 0.

math.RA↗
arXiv2024

Translational hulls of semigroups of endomorphisms of an algebra

We consider the translational hull $\Omega(I)$ of an arbitrary subsemigroup $I$ of an endomorphism monoid $\mathrm{End}(A)$ where $A$ is a universal algebra. We give conditions for every bi-translation of $I$ to be realised by transformations, or by endomorphisms, of $A$. We demonstrate that certain of these conditions are also sufficient to provide natural isomorphisms between the translational hull of $I$ and the idealiser of $I$ within $\mathrm{End}(A)$, which in the case where $I$ is an ideal is simply $\mathrm{End}(A)$. We describe the connection between these conditions and work of Petrich and Gluskin in the context of densely embedded ideals. Where the conditions fail, we develop a methodology to extract information concerning $\Omega(I)$ from the translational hull $\Omega(I/{\approx})$ of a quotient $I/{\approx}$ of $I$. We illustrate these concepts in detail in the cases where $A$ is: a free algebra; an independence algebra; a finite symmetric group.

math.RA↗
arXiv2024

D-inverse constellations

We give an algebraic characterisation of ordered groupoids, namely, we show that there is a categorical isomophism between the category of ordered groupoids and the category of $D$-inverse constellations. Here constellations are partial algebras in the sense that they possess a partial product, and a unary operation $D$. We consider constellations in which elements have a suitable notion of inverse, giving the notion of a D-inverse constellation.

math.CT↗
arXiv2023

Coherency properties for monoids of transformations and partitions

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act itself has a finite presentation; it is weakly right coherent if every finitely generated right ideal of $S$ has a finite presentation. We show that full and partial transformation monoids, symmetric inverse monoids and partition monoids over an infinite set are all weakly right coherent, but that none of them is right coherent. Left coherency and weak left coherency are defined dually, and the corresponding results hold for these properties. In order to prove the non-coherency results, we give a presentation of an inverse semigroup which does not embed into any left or right coherent monoid.

math.RA↗
arXiv2023

On the diameter of semigroups of transformations and partitions

For a semigroup $S$ whose universal right congruence is finitely generated (or, equivalently, a semigroup satisfying the homological finiteness property of being type right-$FP_1$), the right diameter of $S$ is a parameter that expresses how `far apart' elements of $S$ can be from each other, in a certain sense. To be more precise, for each finite generating set $U$ for the universal right congruence on $S,$ we have a metric space $(S,d_U)$ where $d_U(a,b)$ is the minimum length of derivations for $(a,b)$ as a consequence of pairs in $U$; the right diameter of $S$ with respect to $U$ is the diameter of this metric space. The right diameter of $S$ is then the minimum of the set of all right diameters with respect to finite generating sets. We investigate whether various natural infinite semigroups of transformations and partitions have a finitely generated universal right/left congruence, and for those that do, we determine their right/left diameter. Among other results, for an arbitrary infinite set $X$ we prove the following. Each of the monoids of all binary relations on $X,$ of all partial transformations on $X,$ and of all full transformations on $X,$ as well as the partition and partial Brauer monoids on $X,$ have right diameter 1 and left diameter 1. The symmetric inverse monoid on $X$ has right diameter 2 and left diameter 2. The monoid of all injective mappings on $X$ has right diameter 4, and its minimal ideal (called the Baer-Levi semigroup on $X$) has right diameter 3, but neither of these two semigroups has a finitely generated universal left congruence. On the other hand, the semigroup of all surjective mappings on $X$ has left diameter 4, and its minimal ideal has left diameter 2, but neither of these semigroups has a finitely generated universal right congruence.

math.GR↗
arXiv2023

The structure of End($\mathcal{T}_n$)

The full transformation semigroups $\mathcal{T}_n$, where $n\in \mathbb{N}$, consisting of all maps from a set of cardinality $n$ to itself, are arguably the most important family of finite semigroups. This article investigates the endomorphism monoid End($\mathcal{T}_n$) of $\mathcal{T}_n$. The determination of the elements of End($\mathcal{T}_n$) is due Schein and Teclezghi. Surprisingly, the algebraic structure of End($\mathcal{T}_n$) has not been further explored. We describe Green's relations and extended Green's relations on End($\mathcal{T}_n$), and the generalised regularity properties of these monoids. In particular, we prove that $\mathcal{H}=\mathcal{L} \subseteq \mathcal{R}= \mathcal{D}=\mathcal{J}$ (with equality if and only if $n=1$); the idempotents of End($\mathcal{T}_n$) form a band (which is equal to End($\mathcal{T}_n$) if and only if $n=1$) and also the regular elements of End($\mathcal{T}_n$) form a subsemigroup (which is equal to End($\mathcal{T}_n$) if and only if $n\leq 2$). Further, the regular elements of End($\mathcal{T}_n$) are precisely the idempotents together with all endomorphisms of rank greater than $3$. We also provide a presentation for End($\mathcal{T}_n$) with respect to a minimal generating set.

math.RA↗
arXiv2022

Coherency for monoids and purity for their acts

This article examines the three-way relationship between right coherency of a monoid $S$, solutions of equations over $S$-acts, and injectivity properties of $S$-acts. A monoid $S$ is right coherent if every finitely generated subact of every finitely presented (right) $S$-act itself has a finite presentation. Purity properties of an $S$-act $A$ may either be expressed in terms of solutions in $A$ of certain consistent sets of equations over $A$, or in terms of injectivity properties. For example, an $S$-act $A$ is absolutely pure (almost pure) if every finite consistent set of equations over $A$ (in one variable) has a solution in $A$. Equivalently, $A$ is absolutely pure (almost pure) if it is injective with respect to inclusions of finitely generated subacts into finitely presented (monogenic finitely presented) $S$-acts. Our first main result shows that for a right coherent monoid $S$ the classes of almost pure and absolutely pure $S$-acts coincide. Our second main result is that a monoid $S$ is right coherent if and only if the classes of mfp-pure and absolutely pure $S$-acts coincide: an $S$-act is mfp-pure if it is injective with respect to inclusions of finitely presented subacts into monogenic finitely presented $S$-acts. We give specific examples of monoids $S$ that are not right coherent yet are such that the classes of almost pure and absolutely pure $S$-acts coincide. Finally we give a condition on a monoid $S$ for all almost pure $S$-acts to be absolutely pure in terms of finitely presented $S$-acts, their finitely generated subacts, and certain canonical extensions.

math.GR↗
arXiv2022

Semigroups of straight left inverse quotients

Let $Q$ be an inverse semigroup. A subsemigroup $S$ of $Q$ is a left I-order in $Q$ and $Q$ is a semigroup of left I-quotients of $S$ if every element in $Q$ can be written as $a^{-1}b$, where $a, b \in S$ and $a^{-1}$ is the inverse of $a$ in the sense of inverse semigroup theory. If we insist on being able to take $a$ and $b$ to be $\mathcal{R}$-related in $Q$ we say that $S$ is straight in $Q$ and $Q$ is a semigroup of straight left I-quotients of $S$. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of $\mathcal{R}$ and $\mathcal{L}$ from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the $\mathcal{L}$ of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight.

math.RA↗
arXiv2022

Product decompositions of semigroups induced by action pairs

This paper concerns a class of semigroups that arise as products $US$, associated to what we call `action pairs'. Here $U$ and $S$ are subsemigroups of a common monoid and, roughly speaking, $S$ has an action on the monoid completion $U^1$ that is suitably compatible with the product in the over-monoid. The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure a comprehensive understanding of them. We then apply our abstract results to many special cases of interest. The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup $US$ is a quotient of a semidirect product $U\rtimes S$, and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (a.k.a. $E$-unitary) inverse semigroups. The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for $US$ out of presentations for the constituents $U$ and $S$ in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.

math.RA↗
arXiv2022

On minimal ideals in pseudo-finite semigroups

A semigroup $S$ is said to be right pseudo-finite if the universal right congruence can be generated by a finite set $U\subseteq S\times S$, and there is a bound on the length of derivations for an arbitrary pair $(s,t)\in S\times S$ as a consequence of those in $U$. This article explores the existence and nature of a minimal ideal in a right pseudo-finite semigroup. Continuing the theme started in an earlier work by Dandan et al., we show that in several natural classes of monoids, right pseudo-finiteness implies the existence of a completely simple minimal ideal. This is the case for orthodox monoids, completely regular monoids and right reversible monoids, which include all commutative monoids. We also show that certain other conditions imply the existence of a minimal ideal, which need not be completely simple; notably, this is the case for semigroups in which one of the Green's pre-orders $\leq_{\mathcal{L}}$ or $\leq_{\mathcal{J}}$ is left compatible with multiplication. Finally, we establish a number of examples of pseudo-finite monoids without a minimal ideal. We develop an explicit construction that yields such examples with additional desired properties, for instance, regularity or $\mathcal{J}$-triviality.

math.GR↗
arXiv2021

Constellations with range and IS-categories

Constellations are asymmetric generalisations of categories. Although they are not required to possess a notion of range, many natural examples do. These include commonly occurring constellations related to concrete categories (since they model surjective morphisms), and also others arising from quite different sources, including from well-studied classes of semigroups. We show how constellations with a well-behaved range operation are nothing but ordered categories with restrictions. We characterise abstractly those categories that are canonical extensions of constellations with range, as so-called IS-categories. Such categories contain distinguished subcategories of insertions (which are monomorphisms) and surjections (in general different to the epimorphisms) such that each morphism admits a unique factorisation into a surjection followed by an insertion. Most familiar concrete categories are IS-categories, and we show how some of the well-known properties of these categories arise from the fact that they are IS-categories. For appropriate choices of morphisms in each, the category of IS-categories is shown to be equivalent to the category of constellations with range.

math.CT↗