arXiv subjects

James D. Mitchell

Publications and source records attributed to James D. Mitchell.

At least 19 recordsLinked to original sources

arXiv2025

Computing finite index congruences of finitely presented semigroups and monoids

In this paper, we describe an algorithm for computing the left, right, or 2-sided congruences of a finitely presented semigroup or monoid with finitely many classes, and an alternative algorithm when the finitely presented semigroup or monoid is finite. We compare the two algorithms presented with existing algorithms and implementations. The first algorithm is a generalization of Sims' low-index subgroup algorithm for finding the congruences of a monoid. The second algorithm involves determining the distinct principal congruences, and then finding all of their possible joins. Variations of this algorithm have been suggested in numerous contexts by numerous authors. We show how to utilize the theory of relative Green's relations, and a version of Schreier's Lemma for monoids, to reduce the number of principal congruences that must be generated as the first step of this approach. Both of the algorithms described in this paper are implemented in the GAP package Semigroups, and the first algorithm is available in the C++ library libsemigroups and in its python bindings libsemigroups_pybind11.

math.RA↗

arXiv2024

Minimum transformation representations of diagram monoids

We obtain formulae for the minimum transformation degrees of the most well-studied families of finite diagram monoids, including the partition, Brauer, Temperley--Lieb and Motzkin monoids. For example, the partition monoid $P_n$ has degree $1 + \frac{B(n+2)-B(n+1)+B(n)}2$ for $n\geq2$, where these are Bell numbers. The proofs involve constructing explicit faithful representations of the minimum degree, many of which can be realised as (partial) actions on projections.

math.RA↗

arXiv2024

Semirigidity and the enumeration of nilpotent semigroups of index three

There is strong evidence for the belief that `almost all' finite semigroups, whether we consider multiplication operations on a fixed set or their isomorphism classes, are nilpotent of index 3 (3-nilpotent for short). The only known method for counting all semigroups of given order is exhaustive testing, but formulae exist for the numbers of 3-nilpotent ones, and it is also known that `almost all' of these are rigid (have only trivial automorphism). Here we express the number of distinct 3-nilpotent semigroup operations on a fixed set of cardinality $n$ as a sum of Stirling numbers, and provide a new expression for the number of isomorphism classes of 3-nilpotent semigroups of cardinality $n$. We introduce a notion of semirigidity for semigroups (as a generalization of rigidity) and find computationally tractable formulae giving an upper bound for the number of pairwise non-isomorphic semirigid 3-nilpotent semigroups, and thus an improved lower bound for the number of all 3-nilpotent semigroups up to isomorphism. Analogous formulae are also developed for isomorphism classes such as commutative and self-dual semigroups, and for equivalence classes (isomorphic or anti-isomorphic). The method relies on an application of the theory of orbit counting in permutation group actions. Our main results are accompanied by tables containing values of these numbers and bounds up to $n=10$ with computations carried out in GAP (but perfectly feasible well beyond this value of $n$).

math.CO↗

arXiv2024

Short presentations for transformation monoids

By the theorems of Cayley and Vagner-Preston, the full transformation monoids and the symmetric inverse monoids play analogous roles in the theory of monoids and inverse monoids, as the symmetric groups do in the theory of groups. Every presentation for the finite full transformation monoids $T_n$, symmetric inverse monoids $I_n$, and partial transformation monoids $PT_n$ contains a monoid presentation for the symmetric group. In this paper we show that the number of relations required, in addition to those for the symmetric group, for each of these monoids is at least $4$, $3$, and $8$, respectively. We also give presentations for: $T_n$ with $4$ additional relations when $n\geq 7$; for $I_n$ with $3$ additional relations for all $n \geq 3$; and for $PT_n$ with $8$ additional relations for all $n\geq 7$. The presentations for $T_n$ and $I_n$ answer open problems in the literature.

math.GR↗

arXiv2024

Power Quotients of Plactic-like Monoids

In this paper we describe the quotients of several plactic-like monoids by the least congruences containing the relations $a^{σ(a)} = a$ with $σ(a)\ge 2$ for every generator $a$. The starting point for this description is the recent paper of Abram and Reutenauer about the so-called stylic monoid which happens to be the quotient of the plactic monoid by the relations $a^2 = a$ for every letter $a$. The plactic-like monoids considered are the plactic monoid itself, the Chinese monoid, and the sylvester monoid. In each case we describe: a set of normal forms, and the idempotents; and obtain formulae for their size.

math.CO↗

arXiv2024

Computing congruences of finite inverse semigroups

In this paper we present a novel algorithm for computing a congruence on an inverse semigroup from a collection of generating pairs. This algorithm uses a myriad of techniques from the theories of groups, automata, and inverse semigroups. An initial implementation of this algorithm outperforms existing implementations by several orders of magnitude.

math.GR↗

arXiv2023

An algebraic model for inversion and deletion in bacterial genome rearrangement

Inversions, also sometimes called reversals, are a major contributor to variation among bacterial genomes, with studies suggesting that those involving small numbers of regions are more likely than larger inversions. Deletions may arise in bacterial genomes through the same biological mechanism as inversions, and hence a model that incorporates both is desirable. However, while inversion distances between genomes have been well studied, there has yet to be a model which accounts for the combination of both deletions and inversions. To account for both of these operations, we introduce an algebraic model that utilises partial permutations. This leads to an algorithm for calculating the minimum distance to the most recent common ancestor of two bacterial genomes evolving by inversions (of adjacent regions) and deletions. The algebraic model makes the existing short inversion models more complete and realistic by including deletions, and also introduces new algebraic tools into evolutionary distance problems.

math.RA↗

arXiv2023

An explicit algorithm for normal forms in small overlap monoids

We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the $C(4)$ condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in $C(4)$ monoids: the word problem is solvable in $O(\min\{|u|, |v|\})$ time in the size of the input words $u$ and $v$; the uniform word problem for $\langle A|R\rangle$ is solvable in $O(N ^ 2 \min\{|u|, |v|\})$ where $N$ is the sum of the lengths of the words in $R$; and a normal form for any given word $u$ can be found in $O(|u|)$ time. Although Kambites' algorithm for solving the word problem in $C(4)$ monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in $C(4)$ monoids that has time complexity $O(|u| ^ 2)$ for input word $u$, but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in $O(N \min\{|u|, |v|\})$ time.

cs.FL↗

arXiv2023

Properties of Congruence Lattices of Graph Inverse Semigroups

From any directed graph $E$ one can construct the graph inverse semigroup $G(E)$, whose elements, roughly speaking, correspond to paths in $E$. Wang and Luo showed that the congruence lattice $L(G(E))$ of $G(E)$ is upper-semimodular for every graph $E$, but can fail to be lower-semimodular for some $E$. We provide a simple characterisation of the graphs $E$ for which $L(G(E))$ is lower-semimodular. We also describe those $E$ such that $L(G(E))$ is atomistic, and characterise the minimal generating sets for $L(G(E))$ when $E$ is finite and simple.

math.RA↗

arXiv2019

Sets of universal sequences for the symmetric group and analogous semigroups

A universal sequence for a group or semigroup $S$ is a sequence of words $w_1, w_2, \ldots$ such that for any sequence $s_1, s_2, \ldots\in S$, the equations $w_n = s_n$, $n\in \mathbb{N}$, can be solved simultaneously in $S$. For example, Galvin showed that the sequence $(a^{-1}(a^nba^{-n})b^{-1}(a^nb^{-1}a^{-n})ba)_{n\in\mathbb{N}}$ is universal for the symmetric group Sym$(X)$ when $X$ is infinite, and Sierpiński showed that $(a ^ 2 b ^ 3 (abab ^ 3) ^ {n + 1} ab ^ 2 ab ^ 3)_{n\in \mathbb{N}}$ is universal for the monoid $X ^ X$ of functions from the infinite set $X$ to itself. In this paper, we show that under some conditions, the set of universal sequences for the symmetric group on an infinite set $X$ is independent of the cardinality of $X$. More precisely, we show that if $Y$ is any set such that $|Y| \geq |X|$, then every universal sequence for Sym$(X)$ is also universal for Sym$(Y)$. If $|X| > 2 ^ {\aleph_0}$, then the converse also holds. It is shown that an analogue of this theorem holds in the context of inverse semigroups, where the role of the symmetric group is played by the symmetric inverse monoid. In the general context of semigroups, the full transformation monoid $X ^ X$ is the natural analogue of the symmetric group and the symmetric inverse monoid. If $X$ and $Y$ are arbitrary infinite sets, then it is an open question as to whether or not every sequence that is universal for $X ^ X$ is also universal for $Y ^ Y$. However, we obtain a sufficient condition for a sequence to be universal for $X ^ X$ which does not depend on the cardinality of $X$. A large class of sequences satisfy this condition, and hence are universal for $X ^ X$ for every infinite set $X$.

math.GR↗

arXiv2018

Maximal subsemigroups of finite transformation and diagram monoids

We describe and count the maximal subsemigroups of many well-known monoids of transformations and monoids of partitions. More precisely, we find the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer, Jones, and Motzkin monoids; and certain further monoids. Although descriptions of the maximal subsemigroups of some of the aforementioned classes of monoids appear in the literature, we present a unified framework for determining these maximal subsemigroups. This approach is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors. This allows us to concisely present the descriptions of the maximal subsemigroups, and to more clearly see their common features.

math.GR↗

arXiv2018

Automorphism groups of linearly ordered structures and endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers

We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly ordered set $Ω$, there are uncountably many maximal subgroups of $\operatorname{End}(\mathbb{Q},{\leq})$ isomorphic to the automorphism group of $Ω$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as a retract in $(\mathbb{Q},{\leq})$ in terms of topological information concerning $X$. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal subgroup of $\operatorname{End}(\mathbb{Q},{\leq})$, if and only if it is free abelian of finite rank.

math.GR↗

arXiv2018

Congruence lattices of finite diagram monoids

We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I->M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.

math.GR↗

arXiv2017

Structural aspects of semigroups based on digraphs

Given any digraph $D$ without loops or multiple arcs, there is a natural construction of a semigroup $\langle D\rangle$ of transformations. To every arc $(a,b)$ of $D$ is associated the idempotent transformation $(a\to b)$ mapping $a$ to $b$ and fixing all vertices other than $a$. The semigroup $\langle D\rangle$ is generated by the idempotent transformations $(a\to b)$ for all arcs $(a,b)$ of $D$. In this paper, we consider the question of when there is a transformation in $\langle D\rangle$ containing a large cycle, and, for fixed $k\in \mathbb N$, we give a linear time algorithm to verify if $\langle D\rangle$ contains a transformation with a cycle of length $k$. We also classify those digraphs $D$ such that $\langle D\rangle$ has one of the following properties: inverse, completely regular, commutative, simple, 0-simple, a semilattice, a rectangular band, congruence-free, is $\mathscr{K}$-trivial or $\mathscr{K}$-universal where $\mathscr{K}$ is any of Green's $\mathscr{H}$-, $\mathscr{L}$-, $\mathscr{R}$-, or $\mathscr{J}$-relation, and when $\langle D\rangle$ has a left, right, or two-sided zero.

math.CO↗

arXiv2017

Enumerating Transformation Semigroups

We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation semigroups up to degree 4. Classification of these semigroups up to conjugacy, isomorphism and anti-isomorphism, by size and rank, provides a solid base for further investigations of transformation semigroups.

math.GR↗

arXiv2016

Computing Embeddings and Isomorphisms of Finite Semigroups

Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering questions like `What is the minimal number of states to realize a particular computation?' and `Which type of computation is more capable?' translate into the algebraic tasks of constructing isomorphisms and embeddings between semigroups of different representations. The underlying problem is (sub)graph isomorphism, which is computationally difficult in general. We describe variations of backtrack search algorithms that exploit the algebraic properties of semigroups, and we carry out computational experiments to extend our algebraic knowledge. In particular, we report new computational results on transformation semigroups and on the more general family of diagram semigroups. We study the minimal degree representation problem, count distinct embeddings and work on an open problem of embedding into 2-generated subsemigroups.

math.GR↗

arXiv2015

Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.

math.CO↗

arXiv2015

Idempotent rank in the endomorphism monoid of a non-uniform partition

We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.

math.GR↗