arXiv subjects

Victoria Gould

Publications and source records attributed to Victoria Gould.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2020

Coherency and constructions for monoids

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck--Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of $S$ to be finitely generated.

math.GR↗

arXiv2019

Independence algebras, basis algebras and the distributivity condition

Stable basis algebras were introduced by Fountain and Gould and developed in a series of articles. They form a class of universal algebras, extending that of independence algebras. If a stable basis algebra $\mathbb{B}$ of finite rank satisfies the distributivity condition (a condition satisfied by all the previously known examples), it is a reduct of an independence algebra $\mathbb{A}$. Our first aim is to give an example of an independence algebra not satisfying the distributivity condition. Gould showed that if a stable basis algebra $\mathbb{B}$ with the distributivity condition has finite rank, then so does the independence algebra $\mathbb{A}$ of which it is a reduct, and in this case the endomorphism monoid End$(\mathbb{B})$ of $\mathbb{B}$ is a left order in the endomorphism monoid End$(\mathbb{A})$ of $\mathbb{A}$. We complete the picture by determining when End$(\mathbb{B})$ is a right, and hence a two-sided, order in End$(\mathbb{A})$. In fact (for rank at least 2), this happens precisely when every element of End$(\mathbb{A})$ can be written as $α^\sharpβ$ where $α,β\in$ End$(\mathbb{B})$, $α^\sharp$ is the inverse of $α$ in a subgroup of End$(\mathbb{A})$ and $α$ and $β$ have the same kernel. This is equivalent to End$(\mathbb{B})$ being a special kind of left order in End$(\mathbb{A})$ known as straight.

math.RA↗

arXiv2019

Matrix semigroups over semirings

The multiplicative semigroup $M_n(F)$ of $n\times n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ consisting of upper triangular and unitriangular matrices. Our main interest is in the case where $S$ is an idempotent semifield, where we also consider the subsemigroups $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our guiding examples of such $S$ are the 2-element Boolean semiring $\mathbb{B}$ and the tropical semiring $\mathbb{T}$. In the first case, $M_n(\mathbb{B})$ is isomorphic to the semigroup of binary relations on an $n$-element set, and in the second, $M_n(\mathbb{T})$ is the semigroup of $n\times n$ tropical matrices. Il'in has proved that for any semiring $R$ and $n>2$, the semigroup $M_n(R)$ is regular if and only if $R$ is a regular ring. We therefore base our investigations for $M_n(S)$ and its subsemigroups on the analogous but weaker concept of being Fountain (formerly, weakly abundant). These notions are determined by the existence and behaviour of idempotent left and right identities for elements, lying in particular equivalence classes. We show that certain subsemigroups of $M_n(S)$, including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We give a detailed study of a family of Fountain semigroups arising in this way that has particularly interesting and unusual properties.

math.RA↗

arXiv2019

A group-theoretical interpretation of the word problem for free idempotent generated semigroups

The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup $\mathsf{IG}(\mathcal{E})$ - the `free-est' semigroup with a given biordered set $\mathcal{E}$ of idempotents. We show that when $\mathcal{E}$ is finite, the word problem for $\mathsf{IG}(\mathcal{E})$ is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of $\mathsf{IG}(\mathcal{E})$. As an application, we obtain decidability of the word problem for an important class of examples. Also, we prove that for finite $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$ is always a weakly abundant semigroup satisfying the congruence condition.

math.GR↗

arXiv2018

Presentations for singular wreath products

For a monoid $M$ and a subsemigroup $S$ of the full transformation semigroup $T_n$, the wreath product $M\wr S$ is defined to be the semidirect product $M^n\rtimes S$, with the coordinatewise action of $S$ on $M^n$. The full wreath product $M\wr T_n$ is isomorphic to the endomorphism monoid of the free $M$-act on $n$ generators. Here, we are particularly interested in the case that $S=Sing_n$ is the singular part of $T_n$, consisting of all non-invertible transformations. Our main results are presentations for $M\wr Sing_n$ in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that $M\wr Sing_n$ is idempotent generated if and only if the set $M/L$ of $L$-classes of $M$ forms a chain under the usual ordering of $L$-classes, and we give a presentation for $M\wr Sing_n$ in terms of idempotent generators for such a monoid $M$. Among other results, we also give estimates for the minimal size of a generating set for $M\wr Sing_n$, as well as exact values in some cases (including the case that $M$ is finite and $M/L$ is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set.

math.GR↗

arXiv2018

Model-theoretic properties of free, projective, and flat S-acts

This is the second in a series of articles surveying the body of work on the model theory of S-acts over a monoid S. The first concentrated on the theory of regular S-acts. Here we review the material on model-theoretic properties of free, projective, and (strongly, weakly) flat S-acts. We consider questions of axiomatizability, completeness, model completeness, and stability for these classes. Most but not all of the results have already appeared; we remark that the description of those monoids S such that the class of free left S-acts is axiomatizable, is new.

math.LO↗

arXiv2018

$\aleph_0$-categoricity of semigroups

In this paper we initiate the study of $\aleph_0$-categorical semigroups, where a countable semigroup $S$ is $\aleph_0$-categorical if, for any natural number $n$, the action of its group of automorphisms Aut $S$ on $S^n$ has only finitely many orbits. We show that $\aleph_0$-categoricity transfers to certain important substructures such as maximal subgroups and principal factors. We examine the relationship between $\aleph_0$-categoricity and a number of semigroup and monoid constructions, namely direct sums, 0-direct unions, semidirect products and $\mathcal{P}$-semigroups. As a corollary, we determine the $\aleph_0$-categoricity of an $E$-unitary inverse semigroup with finite semilattice of idempotents in terms of that of the maximal group homomorphic image.

math.LO↗

arXiv2016

Embedding in factorisable restriction monoids

Each restriction semigroup is proved to be embeddable in a factorisable restriction monoid, or, equivalently, in an almost factorisable restriction semigroup. It is also established that each restriction semigroup has a proper cover which is embeddable in a semidirect product of a semilattice by a group.

math.RA↗

arXiv2015

Constellations and their relationship with categories

Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories, showing the latter to be special cases of the former. In particular, we characterise those constellations that arise as (sub-)reducts of categories, and show that categories are nothing but two-sided constellations. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation $P$ gives rise to a category $\mathcal{C}(P)$, its canonical extension, in a simplest possible way, and that $P$ is a quotient of $\mathcal{C}(P)$ in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences $δ$ on a given category $K$ (those for which $K\cong \mathcal{C}((K/δ)$), and show that the category of constellations is equivalent to the category of categories equipped with distinguished canonical congruence.

math.CT↗

arXiv2015

Free monoids are coherent

A monoid $S$ is said to be right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. Left coherency is defined dually and $S$ is coherent if it is both right and left coherent. These notions are analogous to those for a ring $R$ (where, of course, $S$-acts are replaced by $R$-modules). Choo, Lam and Luft have shown that free rings are coherent. In this note we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by the first author in 1992.

math.RA↗

arXiv2015

Free idempotent generated semigroups over bands

Free idempotent generated semigroups IG$(E)$, where $E$ is a biordered set, have provided a focus of recent research, the majority of the efforts concentrating on the behaviour of the maximal subgroups. Inspired by an example of Brittenham, Margolis and Meakin, several proofs have been offered that any group occurs as a maximal subgroup of some IG$(E)$, the most recent being that of Dolinka and Ruškuc, who show that $E$ can be taken to be a band. From a result of Easdown, Sapir and Volkov, periodic elements of any IG$(E)$ lie in subgroups. However, little else is known of the `global' properties of IG$(E)$, other than that it need not be regular, even where $E$ is a semilattice. Since its introduction by Fountain in the late 1970s, the study of abundant and related semigroups has given rise to a deep and fruitful research area. The classes of abundant and adequate semigroups extend those of regular and inverse semigroups, respectively, and themselves are contained in the classes of weakly abundant and weakly adequate semigroups. Recent significant developments include the description by Kambites, using birooted labelled trees, of the free semigroups in the quasi-variety of adequate semigroups. Our main result shows that for any band $B$, the semigroup IG$(B)$ is a weakly abundant semigroup and moreover satisfies a natural condition called the congruence condition. We show that if $B$ is a band for which $uv=vu=v$ for all $u,v\in B$ with $BvB\subset BuB$ (a condition certainly satisfied for semilattices), then IG$(B)$ is abundant with solvable word problem. Further, IG$(B)$ is also abundant for a normal band $B$ for which IG$(B)$ satisfies a given technical condition, and we give examples of such $B$. On the other hand, we give an example of a normal band $B$ such that IG$(B)$ is not abundant.

math.RA↗

arXiv2015

Coherency, free inverse monoids and free left ample monoids

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. The corresponding notion for a ring $R$ states that every finitely generated submodule of every finitely presented right $R$-module is finitely presented. For monoids (and rings) right coherency is a finitary property which determines the existence of a model companion of the class of right $S$-acts (right $R$-modules) and hence that the class of existentially closed right $S$-acts (right $R$-modules) is axiomatisable. Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruskuc, have shown that groups, and free monoids, have the same properties. We demonstrate that free inverse monoids do not. Any free inverse monoid contains as a submonoid the free left ample monoid, and indeed the free monoid, on the same set of generators. The main objective of the paper is to show that the free left ample monoid is right coherent. Furthermore, by making use of the same techniques we show that both free inverse and free left ample monoids satisfy $({\bf R})$, $({\bf r)}$, $({\bf L})$ and $({\bf l)}$, conditions arising from the axiomatisability of classes of right $S$-acts and of left $S$-acts.

math.RA↗

arXiv2014

Free idempotent generated semigroups and endomorphism monoids of free $G$-acts

The study of the free idempotent generated semigroup $\mathrm{IG}(E)$ over a biordered set $E$ began with the seminal work of Nambooripad in the 1970s and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Here we study $\mathrm{IG}(E)$ in the case $E$ is the biordered set of a wreath product $G\wr \mathcal{T}_n$, where $G$ is a group and $\mathcal{T}_n$ is the full transformation monoid on $n$ elements. This wreath product is isomorphic to the endomorphism monoid of the free $G$-act $F_n(G)$ on $n$ generators, and this provides us with a convenient approach. We say that the rank of an element of $F_n(G)$ is the minimal number of (free) generators in its image. Let $\varepsilon=\varepsilon^2\in F_n(G).$ For rather straightforward reasons it is known that if $\mathrm{rank}\,\varepsilon =n-1$ (respectively, $n$), then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is free (respectively, trivial). We show that if $\mathrm{rank}\,\varepsilon =r$ where $1\leq r\leq n-2$, then the maximal subgroup of $\mathrm{IG}(E)$ containing $\varepsilon$ is isomorphic to that in $F_n(G)$ and hence to $G\wr \mathcal{S}_r$, where $\mathcal{S}_r$ is the symmetric group on $r$ elements. We have previously shown this result in the case $ r=1$; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case $r=1$ and thus provides another approach to showing that any group occurs as the maximal subgroup of some $\mathrm{IG}(E)$. On the other hand, varying $r$ again and taking $G$ to be trivial, we obtain an alternative proof of the recent result of Gray and Ruškuc for the biordered set of idempotents of $\mathcal{T}_n.$

math.GR↗

arXiv2012

Every group is the maximal subgroup of a naturally occurring free idempotent generated semigroup

Gray and Ruskuc have shown that any group G occurs as the maximal subgroup of some free idempotent generated semigroup IG(E) on a biordered set of idempotents E, thus resolving a long standing open question. Given the group G, they make a careful choice for E and use a certain amount of well developed machinery. Our aim here is to present a short and direct proof of the same result, moreover by using a naturally occuring biordered set. More specifically, for any free G-act F_n(G) of finite rank at least 3, we have that G is a maximal subgroup of IG(E) where E is the biordered set of idempotents of End F_n(G). Note that if G is finite then so is End F_n(G).

math.RA↗

arXiv2010

Inverse semigroups of left I-quotients

We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford's seminal work describing bisimple inverse monoids in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford's work to bisimple inverse semigroups (a step that has previously proved to be awkward). We also put some earlier work on Gantos into a wider and clearer context, and pave the way for further progress.

math.RA↗

arXiv2009

Axiomatisability problems for S-posets

Let C be a class of algebras of a given fixed type t. Associated with the type is a first order language L_t. One can then ask the question, when is the class C axiomatisable by sentences of L_t. In this paper we will be considering axiomatisability problems for classes of left S-posets over a pomonoid S (that is, a monoid S equipped with a partial order compatible with the binary operation). We aim to determine the pomonoids S such that certain categorically defined classes are axiomatisable. The classes we consider are the free S-posets, the projective S-posets and classes arising from flatness properties. Some of these cases have been studied in a recent article by Pervukhin and Stepanova. We present some general strategies to determine axiomatisability, from which their results for the classes of weakly po-flat and po-flat S-posets will follow. We also consider a number of classes not previously examined.

math.RA↗

arXiv2002

Semigroups of left quotients: existence, straightness and locality

A subsemigroup S of a semigroup Q is a local left order in Q if, for every maximal subgroup H of Q, the intersection of S with H is a local left order in the sense of group theory. That is, every q in H can be written as a#b for some a,b in the intersection of S with H, where here a# denotes the group inverse of a in H. On the other hand, S is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be written as c#d where c,d are in S and if, in addition, every element of S that is square cancellable lies in a subgroup of Q. If one also insists that c and d can be chosen to be related by Green's relation R in Q, then S is said to be a straight left order in Q. This paper investigates the close relation between local left orders and straight left orders in a semigroup Q and gives some quite general conditions for a left order S to be straight. In the light of the connection between locality and straightness we give a complete description of straight left orders that improves upon that in our earlier paper.

math.RA↗