A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.
We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup,namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication.The framework embraces many of the well-known classes of (automatic) semigroups: finite monoids, free semigroups, free commutative monoids, trace or divisibility monoids, braid or Artin-Tits or Krammer or Garside monoids, Baumslag-Solitar semigroups, etc.Like plactic monoids or Chinese monoids, some n…
We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which generalise e.g. the context-free or the indexed languages. When $\mathcal{C}$ is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in $\mathcal{C}$ also has word problem in $\mathcal{C}$. This recovers and generalises a recent result by Brough, Cain & Pfeiffer that the class of c…
The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup. We also show that the class of automaton semigroups is closed under the combined operation of 'free product followed by adjoining an identity'. We present an example of a free product of finite semigroups that we conjecture is not an automaton semigroup. Turning to wreath products, we consider two slight generalizations of the concept of an autom…
We present an algorithmic approach to the conjugacy problems in monoids and semigroups, using rewriting systems. There is a class of monoids and semigroups that satisfy the condition that the transposi- tion problem and the left and right conjugacy problem are equivalent. The free monoid and the completely simple semigroups belong to this class. We give a solution to the conjugacy problem for monoids and semigroups in this class that are presented by a complete rewriting system that satisfies some additional conditions.
We prove that every finite semigroup embeds in a finitely presented congruence-free monoid, and pose some questions around the Boone-Higman Conjecture.
Necessary and sufficient conditions are given for the endomorphism monoid of a profinite semigroup to be profinite. A similar result is established for the automorphism group.
In this paper we characterize the monoid congruences of commutative semigroups by the help of the notion of the separator of subsets of semigroups. We show that every monoid congruence of a commutative semigroup S can be constructed by the help of subsets A of S whose separator is not empty.
Let M be a commutative monoid. We provide an explicit first-order formular that defines the variety generated by M in the lattice of commutative semigroup varieties.
A subset A of a semigroup S is called a medial subset of S if xaby is in A if and only if xbay is in A for every elements x, y, a, b of S. In the paper we show how we can construct the commutative monoid congruences of a semigroup S by the help of medial subsets of S.
We present some fundamental results on (possibly nonlinear) algebraic semigroups and monoids. These include a version of Chevalley's structure theorem for irreducible algebraic monoids, and the description of all algebraic semigroup structures on curves and complete varieties.
We develop some new topological tools to study maximal subgroups of free idempotent generated semigroups. As an application, we show that the rank 1 component of the free idempotent generated semigroup of the biordered set of a full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal subgroup isomorphic to the multiplicative subgroup of Q.
We show that the 42-element monoid of all partial order preserving and extensive injections on the 4-element chain is not contained in any variety generated by a finitely based finite $\mathcal{R}$-trivial semigroup. This provides unified proofs for several known facts and leads to a bunch of new results on the Finite Basis Problem for finite $\mathcal{R}$- and $\mathcal{J}$-trivial semigroups.
On the semigroup of injective monoid endomorphisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three element family $\mathscr{F}^3$ of inductive nonempty subsets of $ω$
This paper studies the classes of semigoups and monoids with context-free and deterministic context-free word problem. First, some examples are exhibited to clarify the relationship between these classes and their connection with the notions of word-hyperbolicity and automaticity. Second, a study is made of whether these classes are closed under applying certain semigroup constructions, including direct products and free products, or under regressing from the results of such constructions to the original semigroup(s) or monoid(s).
On the semigroup of monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with a two-element family $\mathscr{F}$ of inductive nonempty subsets of $ω$
The main result of the paper proves that the monoid of all 2 x 2 tropical matrices satisfies a non-trivial semigroup identity, unlike the case of matrices over infinite fields. We exploit a strong connection between the monoid of all 2 x 2 tropical matrices and the bicyclic monoid to show that a variant of Adian's identity for the bicyclic monoid holds in our case.We use generic matrices of tropical polynomials to prove our result.
We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to …
Recently, Gray and Ruskuc (arXiv:1101.1833) proved that if e is a rank k idempotent transformation of the set {1,...,n} to itself and k<=n-2, then the maximal subgroup of the free idempotent generated semigroup over the full transformation monoid T_n containing e is isomorphic to the symmetric group S_k. We prove that the same holds when T_n is replaced by PT_n, the full monoid of partial transformations on {1,...,n}.
Let T_n be the full transformation semigroup of all mappings from the set {1,...,n} to itself under composition. Let E = E(T_n) denote the set of idempotents of T_n and let e be an arbitrary idempotent satisfying |im(e)|=r < n-1. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group S_r.
The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.
We show that the rank r component of the free idempotent generated semigroup of the biordered set of the full linear monoid of n x n matrices over a division ring Q has maximal subgroup isomorphic to the general linear group GL_r(Q), where n and r are positive integers with r < n/3.
This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…
We introduce an algebraic extension $\boldsymbol{B}_ω^{\mathscr{F}}$ of the bicyclic monoid for an arbitrary $ω$-closed family $\mathscr{F}$ subsets of $ω$ which generalizes the bicyclic monoid, the countable semigroup of matrix units and some other combinatorial inverse semigroups. It is proved that $\boldsymbol{B}_ω^{\mathscr{F}}$ is a combinatorial inverse semigroup and Green's relations, the natural partial order on $\boldsymbol{B}_ω^{\mathscr{F}}$, and its set of idempotents are described. We provide criteria of simplicity, $0$-simplicity, bisimplicity, $0$-bisimplicity of the semigroup $…
We study groupoids and semigroup C*-algebras arising from graphs of monoids, in the setting of right LCM monoids. First, we establish a general criterion when a graph of monoids gives rise to a submonoid of the fundamental group which is right LCM. Moreover, we carry out a detailed analysis of structural properties of semigroup C*-algebras arising from graphs of monoids, including closed invariant subspaces and topological freeness of the groupoids as well as ideal structure, nuclearity and K-theory of the semigroup C*-algebras. As an application, we construct families of pairwise non-conjugat…
We curry the elementary arithmetic operations of addition and multiplication to give monotone injections on N, and describe & study the inverse monoids that arise from also considering their generalised inverses. This leads to well-known classic inverse monoids, as well as a novel inverse monoid (the 'arithmetic inverse monoid' A) that generalises these in a natural number-theoretic manner. Based on this, we interpret classic inverse semigroup theoretic concepts arithmetically, and vice versa. Composition and normal forms within A are based on the Chinese remainder theorem, and a minimal gener…
On the semigroup of injective monoid endomor\-phisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}}$ with the two-elements family $\mathscr{F}$ of inductive nonempty subsets of $ω$
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) …
Let $M$ be a cancellative monoid. It is known~\cite{Ta54} that if $M$ is left amenable then the monoid ring $K[M]$ satisfies Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring. In~\cite{Don10} Donnelly shows that a partial converse to this statement is true. Namely, if the monoid $\mathbb Z^{+}[M]$ of all elements of $\mathbb Z[M]$ with positive coefficients has nonzero common right multiples, then $M$ is left amenable. He asks whether the converse is true for this particular statement. We show that the converse is false even for the case of gro…
For a semigroup $S$, the covering number of $S$ with respect to semigroups, $σ_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an invers…
We study algebraic structure of the $λ$-polycyclic monoid $P_λ$ and its topologizations. We show that the $λ$-polycyclic monoid for an infinite cardinal $λ\geqslant 2$ has similar algebraic properties so has the polycyclic monoid $P_n$ with finitely many $n\geqslant 2$ generators. In particular we prove that for every infinite cardinal $λ$ the polycyclic monoid $P_λ$ is a congruence-free combinatorial $0$-bisimple $0$-$E$-unitary inverse semigroup. Also we show that every non-zero element $x$ is an isolated point in $(P_λ,τ)$ for every Hausdorff topology $τ$ on $P_λ$, such that $(P_λ,τ)$ is a …
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.
For an element a in a semigroup S the local subsemigroup of S with respect to a is the subsemigroup aSa of S and the variant of S with respect to a is a semigroup with underlying set S with a sandwich operation xy = xay for all x, y in S. In this paper we discuss the structures of local subsemigroups of full transformation semigroups and symmetric inverse monoids. It is also shown that the set of all local subsemigroups of finite symmetric inverse monoids and the set of all variants of all finite symmetric inverse monoids are same upto isomorphism.
Consider an algebraic semigroup $S$ and its closed subscheme of idempotents, $E(S)$. When $S$ is commutative, we show that $E(S)$ is finite and reduced; if in addition $S$ is irreducible, then $E(S)$ is contained in a smallest closed irreducible subsemigroup of $S$, and this subsemigroup is an affine toric variety. It follows that $E(S)$ (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when $S$ is an irreducible algebraic monoid, we show that $E(S)$ is smooth, and its connected components are conjugacy classes of the unit group.
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 orthodo…
The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open questio…
Topological embeddings into transformation monoids
P.J. Šafárik University in Košice（科希策帕·约·沙法里克大学）；TU Wien（维也纳工业大学）；Binghamton University（宾汉姆顿大学）；University of St Andrews（圣安德鲁斯大学）；University of Hertfordshire（赫特福德大学）
This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite gro…
In 2010, Izhakian & Margolis proved that the bicyclic monoid admits a representation as a semigroup of upper triangular tropical matrices. We extend this result by classifying all one-relation monoids which admit such representations. We also classify, with two exceptions, all one-relation monoids which admit representations as semigroups of tropical matrices.
We present a construction for the holomorph of an inverse semigroup, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that preserve the ternary heap operation on an inverse semigroup: for groups these two constructions coincide. We present detailed calculations for semilattices of groups and for the polycyclic monoids.
With each semigroup one can associate a partial algebra, called the biordered set, which captures important algebraic and geometric features of the structure of idempotents of that semigroup. For a biordered set $\mathcal{E}$, one can construct the free idempotent-generated semigroup over $\mathcal{E}$, $\mathsf{IG}(\mathcal{E})$, which is the free-est semigroup (in a definite categorical sense) whose biorder of idempotents is isomorphic to $\mathcal{E}$. Studies of these intriguing objects have been recently focusing on their particular aspects, such as maximal subgroups, the word problem, et…
In this paper we study the semigroup $\mathscr{I}^{\infty}_λ$ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality $λ$. We describe the Green relations on $\mathscr{I}^{\infty}_λ$, all (two-sided) ideals and all congruences of the semigroup $\mathscr{I}^{\infty}_λ$. We prove that every Hausdorff hereditary Baire topology $τ$ on $\mathscr{I}^{\infty}_ω$ such that $(\mathscr{I}^{\infty}_ω,τ)$ is a semitopological semigroup is discrete and describe the closure of the discrete semigroup $\mathscr{I}^{\infty}_λ$ in a topological semigroup. Also we show that …
We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid $\mathcal{K}_n$ are nonfinitely based for each $n\ge 3$. This result holds also for the case when $\mathcal{K}_n$ is considered as an involution semigroup under either of its natural involutions.
We study algebraic properties of the Brandt $λ^0$-extensions of monoids with zero and non-trivial homomorphisms between the Brandt $λ^0$-extensions of monoids with zero. We introduce finite, compact topological Brandt $λ^0$-extensions of topological semigroups and countably compact topological Brandt $λ^0$-extensions of topological inverse semigroups in the class of topological inverse semigroups and establish the structure of such extensions and non-trivial continuous homomorphisms between such topological Brandt $λ^0$-extensions of topological monoids with zero. We also describe a category w…
In this paper we study the semigroup $I_\infty^\dnearrow(N)$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $I_\infty^\dnearrow(N)$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $τ$ on $I_\infty^\dnearrow(N)$ such that $(I_\infty^\dnearrow(N),τ)$ is a semitopological semigroup is discrete, describe the closure of $(I_\infty^\dnearrow(N),τ)$ in a…
