arXiv subjects
Publications and source records attributed to J. D. Mitchell.
In this short note we count the finite semirings up to isomorphism, and up to isomorphism or anti-isomorphism for some small values of $n$; for which we utilise the existing library of small semigroups in the GAP package Smallsemi.
We investigate topologies on groups which arise naturally from their algebraic structure, including the Frech\'et-Markov, Hausdorff-Markov, and various kinds of Zariski topologies. Answering a question by Dikranjan and Toller, we show that there exists a countable abelian group in which no bounded version of the Zariski topology coincides with the full Zariski topology. Complementing a recent result by Goffer and Greenfeld, we show that on any group with no algebraicity the semigroup Zariski topology is hyperconnected and hence, in many cases, is distinct from the group Zariski topology. Finally, we show that on the symmetric groups, the semigroup Hausdorff-Markov topology coincides with the topology of pointwise convergence.
In this paper we provide an account of the Todd-Coxeter algorithm for computing congruences on semigroups and monoids. We also give a novel description of an analogue for semigroups of the so-called Felsch strategy from the Todd-Coxeter algorithm for groups.
The height of a poset $P$ is the supremum of the cardinalities of chains in $P$. The exact formula for the height of the subgroup lattice of the symmetric group $\mathcal{S}_n$ is known, as is an accurate asymptotic formula for the height of the subsemigroup lattice of the full transformation monoid $\mathcal{T}_n$. Motivated by the related question of determining the heights of the lattices of left- and right congruences of $\mathcal{T}_n$, we develop a general method for computing the heights of lattices of both one- and two-sided congruences for semigroups. We apply this theory to obtain exact height formulae for several monoids of transformations, matrices and partitions, including: the full transformation monoid $\mathcal{T}_n$, the partial transformation monoid $\mathcal{PT}_n$, the symmetric inverse monoid $\mathcal{I}_n$, the monoid of order-preserving transformations $\mathcal{O}_n$, the full matrix monoid $\mathcal{M}(n,q)$, the partition monoid $\mathcal{P}_n$, the Brauer monoid $\mathcal{B}_n$ and the Temperley-Lieb monoid $\mathcal{TL}_n$.
In this paper we explore the extent to which the algebraic structure of a monoid $M$ determines the topologies on $M$ that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids. If $M$ is a topological monoid such that every homomorphism from $M$ to a second countable topological monoid $N$ is continuous, then we say that $M$ has \emph{automatic continuity}. We show that many well-known monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid $\mathbb{N}^\mathbb{N}$; the full binary relation monoid $B_{\mathbb{N}}$; the partial transformation monoid $P_{\mathbb{N}}$; the symmetric inverse monoid $I_{\mathbb{N}}$; the monoid Inj$(\mathbb{N})$ consisting of the injective functions on $\mathbb{N}$; and the monoid $C(2^{\mathbb{N}})$ of continuous functions on the Cantor set. We show that the pointwise topology on $\mathbb{N}^\mathbb{N}$, and its analogue on $P_{\mathbb{N}}$, are the unique Polish semigroup topologies on these monoids. The compact-open topology is the unique Polish semigroup topology on $C(2^\mathbb{N})$ and $C([0, 1]^\mathbb{N})$. There are at least 3 Polish semigroup topologies on $I_{\mathbb{N}}$, but a unique Polish inverse semigroup topology. There are no Polish semigroup topologies $B_{\mathbb{N}}$ nor on the partitions monoids. At the other extreme, Inj$(\mathbb{N})$ and the monoid Surj$(\mathbb{N})$ of all surjective functions on $\mathbb{N}$ each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on $\mathbb{N}^\mathbb{N}$, $P_{\mathbb{N}}$, and Inj$(\mathbb{N})$ coincide with the pointwise topology; and we characterise the Zariski topology on $B_{\mathbb{N}}$. In Section 7: clones.
We present efficient computational solutions to the problems of checking equality, performing multiplication, and computing minimal representatives of elements of free bands. A band is any semigroup satisfying the identity $x ^ 2 \approx x$ and the free band $\operatorname{FB}(k)$ is the free object in the variety of $k$-generated bands. Radoszewski and Rytter developed a linear time algorithm for checking whether two words represent the same element of a free band. In this paper we describe an alternate linear time algorithm for checking the same problem. The algorithm we present utilises a representation of words as synchronous deterministic transducers that lend themselves to efficient (quadratic in the size of the alphabet) multiplication in the free band. This representation also provides a means of finding the short-lex least word representing a given free band element with quadratic complexity.
In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree $n$ for $n > 0$, equals the sum of the number of cyclic subgroups of the symmetric groups on $1$ to $n$ points. We also prove an analogous statement for monogenic subsemigroups of the finite full transformation monoids, as well as monogenic inverse submonoids and subsemigroups of the finite symmetric inverse monoids.
In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid $\mathbb{N} ^ \mathbb{N}$ or the symmetric inverse monoid $I_{\mathbb{N}}$ with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into $\mathbb{N} ^ \mathbb{N}$ and belong to any of the following classes: commutative semigroups; compact semigroups; groups; and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and $I_{\mathbb{N}}$. We construct several examples of countable Polish topological semigroups that do not embed into $\mathbb{N} ^ \mathbb{N}$, which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of $\mathbb{N}^\mathbb{N}$. The former complements recent works of Banakh et al.
In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid $\operatorname{End}(\mathbb{A})$ of a countable relational structure $\mathbb{A}$. As applications, we show that the endomorphism monoids of several well-known relational structures, including the random graph, the random directed graph, and the random partial order, possess a unique Polish semigroup topology. In every case this unique topology is the subspace topology induced by the usual topology on the Baire space $\mathbb{N} ^ \mathbb{N}$. We also show that many of these structures have the property that every homomorphism from their endomorphism monoid to a second countable topological semigroup is continuous; referred to as automatic continuity. Many of the results about endomorphism monoids are extended to clones of polymorphisms on the same structures.
In this paper, we determine minimal generating sets for several well-known monoids of matrices over semirings. In particular, we find minimal generating sets for the monoids consisting of: all $n\times n$ boolean matrices when $n\leq 8$; the $n\times n$ boolean matrices containing the identity matrix (the reflexive boolean matrices) when $n\leq 7$; the $n\times n$ boolean matrices containing a permutation (the Hall matrices) when $n \leq 8$; the upper, and lower, triangular boolean matrices of every dimension; the $2 \times 2$ matrices over the semiring $\mathbb{N} \cup \{-\infty\}$ with addition $\oplus$ defined by $x\oplus y = \max(x, y)$ and multiplication $\otimes$ given by $x\otimes y = x + y$ (the max-plus semiring); the $2\times 2$ matrices over any quotient of the max-plus semiring by the congruence generated by $t = t + 1$ where $t\in \mathbb{N}$; the $2\times 2$ matrices over the min-plus semiring and its finite quotients by the congruences generated by $t = t + 1$ for all $t\in \mathbb{N}$; and the $n \times n$ matrices over $\mathbb{Z} / n\mathbb{Z}$ relative to their group of units.
In this short note, we describe generating sets for the monoids of consisting of all $2 \times 2$ matrices over certain finite tropical semirings.
We investigate semigroup topologies on the full transformation monoid T(X) of an infinite set X. We show that the standard pointwise topology is the weakest Hausdorff semigroup topology on T(X), show that the pointwise topology is the unique Hausdorff semigroup topology on T(X) that induces the pointwise topology on the group of all permutations of X, and construct |X| distinct Hausdorff semigroup topologies on T(X). In the case where X is countable, we prove that the pointwise topology is the only Polish semigroup topology on T(X). We also show that every separable semigroup topology on T(X) is perfect, describe the compact sets in an arbitrary Hausdorff semigroup topology on T(X), and show that there are no locally compact perfect Hausdorff semigroup topologies on T(X) when |X| has uncountable cofinality.
Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently compute its size and Green's relations, test membership, factorize elements over the generators, find the semigroup generated by the given subsemigroup and any collection of additional elements, calculate the partial order of the $\mathscr{D}$-classes, test regularity, and determine the idempotents. This is achieved by representing the given subsemigroup without exhaustively enumerating its elements. It is also possible to compute the Green's classes of an element of such a subsemigroup without determining the global structure of the semigroup.
A proper subsemigroup of a semigroup is maximal if it is not contained in any other proper subsemigroup. A maximal subsemigroup of a finite semigroup has one of a small number of forms, as described in a paper of Graham, Graham, and Rhodes. Determining which of these forms arise in a given finite semigroup is difficult, and no practical mechanism for doing so appears in the literature. We present an algorithm for computing the maximal subsemigroups of a finite semigroup given knowledge of its Green's structure, and the ability to determine maximal subgroups of certain subgroups. For a finite semigroup $S$ represented by a generating set $X$, in many examples, if it is practical to compute the Green's structure of $S$ from $X$, then it is also practical to find the maximal subsemigroups of $S$ using the algorithm we present. The generating set $X$ for $S$ may consist, for example, of transformations, or partial permutations, of a finite set, or of matrices over a semiring. In such examples, the time taken to determine the Green's structure of $S$ is comparable to that taken to find the maximal subsemigroups. Certain aspects of the problem of finding maximal subsemigroups reduce to other well-known computational problems, such as finding all maximal cliques in a graph and computing the maximal subgroups in a group. The algorithm presented comprises two parts. One part relates to computing the maximal subsemigroups of a special class of semigroups, known as Rees 0-matrix semigroups. The other part involves a careful analysis of certain graphs associated to the semigroup $S$, which, roughly speaking, capture the essential information about the action of $S$ on its $\mathscr{J}$-classes.
In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup. If $U$ is any semigroup, and $A$ is a subset of $U$, then we denote by $\langle A\rangle$ the least subsemigroup of $U$ containing $A$. If $B$ is any other subset of $U$, then, roughly speaking, the first algorithm we present describes how to use any information about $\langle A\rangle$, that has been found using the Froidure-Pin Algorithm, to compute the semigroup $\langle A\cup B\rangle$. More precisely, we describe the data structure for a finite semigroup $S$ given by Froidure and Pin, and how to obtain such a data structure for $\langle A\cup B\rangle$ from that for $\langle A\rangle$. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm.
A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random graph; infinite disjoint unions of complete graphs $K_n$ with $n\in \mathbb{N}$ vertices; the $K_n$-free graphs; finite unions of the infinite complete graph $K_ω$; and duals of such graphs. The groups $\operatorname{Aut}(Γ)$ of automorphisms of such graphs $Γ$ have a natural topology, which is compatible with multiplication and inversion, i.e.\ the groups $\operatorname{Aut}(Γ)$ are topological groups. We consider the problem of finding minimally generated dense subgroups of the groups $\operatorname{Aut}(Γ)$ where $Γ$ is ultrahomogeneous. We show that if $Γ$ is ultrahomogeneous, then $\operatorname{Aut}(Γ)$ has 2-generated dense subgroups, and that under certain conditions given $f \in \operatorname{Aut}(Γ)$ there exists $g\in \operatorname{Aut}(Γ)$ such that the subgroup generated by $f$ and $g$ is dense. We also show that, roughly speaking, $g$ can be chosen with a high degree of freedom. For example, if $Γ$ is either an infinite disjoint unions of $K_n$ or a finite union of $K_ω$, then $g$ can be chosen to have any given finite set of orbit representatives.
To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, Cuntz-Krieger $C^*$-algebras, and Toeplitz $C^*$-algebras. We investigate topologies that turn $G(E)$ into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, $G(E)\setminus \{0\}$ must be discrete for any directed graph $E$. On the other hand, $G(E)$ need not be discrete in a Hausdorff semigroup topology, and for certain graphs $E$, $G(E)$ admits a $T_1$ semigroup topology in which $G(E)\setminus \{0\}$ is not discrete. We also describe, in various situations, the algebraic structure and possible cardinality of the closure of $G(E)$ in larger topological semigroups.
In this paper, we consider the group Aut$(\mathbb{Q}, \leq)$ of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut$(\mathbb{Q}, \leq)$ is contained in an $N$-generated subgroup of Aut$(\mathbb{Q}, \leq)$ for some fixed $N\in\mathbb{N}$. We show that the least such $N$ is $2$. Moreover, for every countable subset of Aut$(\mathbb{Q}, \leq)$, we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that $a$ and $b$ freely generate the free semigroup $\{a,b\}^+$ consisting of the non-empty words over $a$ and $b$. Then we show that there exists a sequence of words $w_1, w_2,\ldots$ over $\{a,b\}$ such that for every sequence $f_1, f_2, \ldots\in\,$Aut$(\mathbb{Q}, \leq)$ there is a homomorphism $ϕ:\{a,b\}^{+}\to$ Aut$(\mathbb{Q},\leq)$ where $(w_i)ϕ=f_i$ for every $i$. As a corollary to the main theorem in this paper, we obtain a result of Droste and Holland showing that the strong cofinality of Aut$(\mathbb{Q}, \leq)$ is uncountable, or equivalently that Aut$(\mathbb{Q}, \leq)$ has uncountable cofinality and Bergman's property.