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J. D. Mitchell

Publications and source records attributed to J. D. Mitchell.

28 records · Page 2Linked to original sources
arXiv2016

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph $D$ on $n$ vertices (with $n\ge2$), there is a natural construction of a semigroup $\langle D\rangle$ associated with $D$. For any edge $(a,b)$ of $D$, let $a\to b$ be the idempotent of defect $1$ mapping $a$ to $b$ and fixing all vertices other than $a$; then define $\langle D\rangle$ to be the semigroup $\langle a\to b:(a,b)\in E(D)\rangle$. For $α\in \langle D \rangle$, let $\ell(D,α)$ be the minimal length of a word in $E(D)$ expressing $α$. When $D=K_n$ is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate $\ell(K_n,α)$, for any $α\in \langle K_n \rangle = \text{Sing}_n$; however, no analogous nontrivial results are known when $D \neq K_n$. In this paper, we characterise all simple digraphs $D$ such that either $\ell(D,α)$ is equal to Howie-Iwahori's formula for all $α\in \langle D \rangle$, or $\ell(D,α) = n - \text{fix}(α)$ for all $α\in \langle D \rangle$, or $\ell(D,α) = n - \text{rk}(α)$ for all $α\in \langle D \rangle$. When $D$ is an acyclic digraph and $α\in \langle D \rangle$, we find a tight upper bound for $\ell(D,α)$. Finally, we study the case when $D$ is a strong tournament (which corresponds to a smallest generating set of idempotents of defect $1$ of $\text{Sing}_n$), and we propose some conjectures.

math.GR↗
arXiv2015

The Structure of a Graph Inverse Semigroup

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.

math.GR↗
arXiv2014

Ends of Semigroups

We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.

math.GR↗
arXiv2014

The rank of the semigroup of transformations stabilising a partition of a finite set

Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a full transformation $f:X\to X$ preserves (or stabilizes) the partition $\mathcal{P}$ if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that $Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the semigroup of all full transformations of $X$ that preserve the partition $\mathcal{P}$. In 2005 Huisheng found an upper bound for the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to completely solve Hisheng's conjecture. The goal of this paper is to solve the much more complex problem of finding the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem. The paper ends with a number of problems for experts in group and semigroup theories.

math.GR↗
arXiv2013

A finite interval in the subsemigroup lattice of the full transformation monoid

In this paper we describe a portion of the subsemigroup lattice of the \emph{full transformation semigroup} $Ω^Ω$, which consists of all mappings on the countable infinite set $Ω$. Gavrilov showed that there are five maximal subsemigroups of $Ω^Ω$ containing the symmetric group $\sym(Ω)$. The portion of the subsemigroup lattice of $Ω^Ω$ which we describe is that between the intersection of these five maximal subsemigroups and $Ω^Ω$. We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{\aleph_0}}$ subsemigroups between $\sym(Ω)$ and $Ω^Ω$.

math.GR↗
arXiv2013

Maximal subsemigroups of the semigroup of all mappings on an infinite set

In this paper we classify the maximal subsemigroups of the \emph{full transformation semigroup} $Ω^Ω$, which consists of all mappings on the infinite set $Ω$, containing certain subgroups of the symmetric group $\sym(Ω)$ on $Ω$. In 1965 Gavrilov showed that there are five maximal subsemigroups of $Ω^Ω$ containing $\sym(Ω)$ when $Ω$ is countable and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary cardinality. We classify the maximal subsemigroups of $Ω^Ω$ on a set $Ω$ of arbitrary infinite cardinality containing one of the following subgroups of $\sym(Ω)$: the pointwise stabiliser of a non-empty finite subset of $Ω$, the stabiliser of an ultrafilter on $Ω$, or the stabiliser of a partition of $Ω$ into finitely many subsets of equal cardinality. If $G$ is any of these subgroups, then we deduce a characterisation of the mappings $f,g\in Ω^Ω$ such that the semigroup generated by $G\cup \{f,g\}$ equals $Ω^Ω$.

math.RA↗
arXiv2013

Ideals and finiteness conditions for subsemigroups

In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub- or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D=J and minimal conditions on ideals.

math.GR↗
arXiv2012

The Bergman-Shelah Preorder on Transformation Semigroups

Let $\nat^\nat$ be the semigroup of all mappings on the natural numbers $\nat$, and let $U$ and $V$ be subsets of $\nat^\nat$. We write $U\preccurlyeq V$ if there exists a countable subset $C$ of $\nat^\nat$ such that $U$ is contained in the subsemigroup generated by $V$ and $C$. We give several results about the structure of the preorder $\preccurlyeq$. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder $\preccurlyeq$ is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on $\nat$. The results in this paper suggest that the preorder on subsemigroups of $\nat^\nat$ is much more complicated than that on subgroups of the symmetric group.

math.LO↗
arXiv2009

Groups that together with any transformation generate regular semigroups or idempotent generated semigroups

Let $a$ be a non-invertible transformation of a finite set and let $G$ be a group of permutations on that same set. Then $\genset{G, a}\setminus G$ is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by $G$ and $a$. Likewise, the conjugates $a^g=g^{-1}ag$ of $a$ by elements $g\in G$ generate a semigroup denoted $\genset{a^g | g\in G}$. We classify the finite permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G,a}$, $\genset{G, a}\setminus G$, and $\genset{a^g | g\in G}$ are regular for all transformations of $X$. We also classify the permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G, a}\setminus G$ and $\genset{a^g | g\in G}$ are generated by their idempotents for all non-invertible transformations of $X$.

math.GR↗