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, etc. In 2012, Gray and Ruškuc pointed out that a more detailed investigation into the structure of the free idempotent-generated semigroup over the biorder of $\mathcal{T}_n$, the full transformation monoid over an $n$-element set, might be worth pursuing. In 2019, together with Gould and Yang, the present author showed that the word problem for $\mathsf{IG}(\mathcal{E}_{\mathcal{T}_n})$ is algorithmically soluble. In a recent work by the author, it was showed that, for a wide class of biorders $\mathcal{E}$, the algorithmic solution of the word problem revolves around the so-called vertex groups, which arise as certain subgroups of direct products of pairs of maximal subgroups of $\mathsf{IG}(\mathcal{E})$. In this paper we determine these vertex groups for the case when $\mathcal{E}$ is the biorder of idempotents of $\mathcal{T}_n$.
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.
We characterise the elements of the (maximum) idempotent generated subsemigroup of the Kauffman monoid in terms of combinatorial data associated to certain normal forms. We also calculate the smallest size of a generating set and idempotent generating set.
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.
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.
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.
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.
We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianised kernel of this idempo…
We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham--Houghton graphs. We show that each proper ideal of the partition monoid P_n is an idempotent generated semigroup, and obtain a formula for the minimal number of elements (and the minimal number of idempotent elements) needed to generate these semigroups. In particular, we show that these two numbers, which are called the rank and idempotent rank (respectively) of the semigroup, are equal to each other, and …
We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships between parameters associated to Brauer graphs. We calculate the relative ranks of each monoid modulo any other such monoid it may contain, and then apply these results to determine the Sierpinski rank of each monoid, and ascertain which ones have the semigroup Bergman property. We also make some fundamental observations about idempotents and units in arbitrar…
Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal P)$ denote the set of all transformations of the finite set $X$ preserving a uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where $m,n\geq2$. We enumerate the idempotents of $\mathcal T(X,\mathcal P)$, and describe the subsemigroup $S=\langle E\rangle$ generated by the idempotents $E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1…
We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated su…
We investigate the structure of the twisted Brauer monoid $\mathcal B_n^τ$, comparing and contrasting it to the structure of the (untwisted) Brauer monoid $\mathcal B_n$. We characterise Green's relations and pre-orders on $\mathcal B_n^τ$, describe the lattice of ideals, and give necessary and sufficient conditions for an ideal to be idempotent-generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent-generated, its…
The algebraic variety defined by the idempotents of an incidence monoid is investigated. Its irreducible components are determined. The intersection with an antichain submonoid is shown to be the union of these irreducible components. The antichain monoids of bipartite posets are shown to be orthodox semigroups. The Green's relations are explicitly determined, and applications to conjugacy problems are described. In particular, it is shown that two elements in the antichain monoid are primarily conjugate in the monoid if and only if they belong to the same $\mathcal{J}$-class and their multipl…
In this paper we present a new embedding of a semigroup into a semiband (idempotent-generated semigroup) of depth 4 (every element is the product of 4 idempotents) using a semidirect product construction. Our embedding does not assume that S is a monoid (although it assumes a weaker condition), and works also for (non-monoid) regular semigroups. In fact, this semidirect product is particularly useful for regular semigroups since we can defined another embedding for these semigroups into a smaller semiband of depth 2. We shall then compare our construction with other known embeddings, and we sh…
For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) category of monoids, $\mathbb G$, $\mathbb E$, $\mathbb G_L$ and $\mathbb G_R$ are all (monoidal) functors $\mathcal M\to\mathcal M$. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We sh…
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 ap…
We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various $\mathscr R$-, $\mathscr L$- and $\mathscr D$-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
The compressed word problem for a finitely generated monoid M asks whether two given compressed words over the generators of M represent the same element of M. For string compression, straight-line programs, i.e., context-free grammars that generate a single string, are used in this paper. It is shown that the compressed word problem for a free inverse monoid of finite rank at least two is complete for Pi^p_2 (second universal level of the polynomial time hierarchy). Moreover, it is shown that there exists a fixed finite idempotent presentation (i.e., a finite set of relations involving idempo…
We show that every finite inverse monoid has an idempotent-separating cover by a finite F-inverse monoid. This provides a positive answer to a conjecture of Henckell and Rhodes.
We prove that the class of closed subgroups of free profinite monoids is precisely the class of projective profinite groups. In particular, the profinite groups associated to minimal symbolic dynamical systems by Almeida are projective. Our result answers a question raised by Lubotzky during the lecture of Almeida at the Fields Workshop on Profinite Groups and Applications, Carleton University, August 2005. We also prove that any finite subsemigroup of a free profinite monoid consists of idempotents.
We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and stro…
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid $L_4^1$ is non-finitely based. The monoid $L_4^1$ was the only unsolved case in the finite basis problem for Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new …
Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids
Centre for Research in Mathematics and Data Science, Western Sydney University（西悉尼大学数学与数据科学研究中心）；School of Engineering, Mathematics and Physics, University of East Anglia（东英吉利大学工程、数学与物理学院）；Mathematical Institute, School of Mathematics and Statistics, University of St Andrews（圣安德鲁斯大学数学与统计学院数学研究所）
We define a subsemigroup $S_n$ of the rook monoid $R_n$ and investigate its properties. To do this, we represent the nonzero elements of $S_n$ (which are $n\times n$ matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that $S_n$ consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, and compute nilpotency indexes. Furthermore, we give a necessary and sufficient condition for the $j$…
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 the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if $\mathcal{C}$ is a reversal-closed super-$\operatorname{AFL}$, as defined by Greibach, then $M$ has word problem in $\mathcal{C}$ if and only if its compressed left monoid $L(M)$ has word problem in $\mathcal{C}$. As a special case, we may take $\mathcal{C}$ to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finall…
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 ge…
We study the semigroup $\mathscr{I}^{\mathrm{cf}}_λ$ of injective partial cofinite selfmaps of an infinite cardinal $λ$. We show that $\mathscr{I}^{\mathrm{cf}}_λ$ is a bisimple inverse semigroup and each chain of idempotents in $\mathscr{I}^{\mathrm{cf}}_λ$ is contained in a bicyclic subsemigroup of $\mathscr{I}^{\mathrm{cf}}_λ$, we describe the Green relations on $\mathscr{I}^{\mathrm{cf}}_λ$ and we prove that every non-trivial congruence on $\mathscr{I}^{\mathrm{cf}}_λ$ is a group congruence. Also, we describe the structure of the quotient semigroup $\mathscr{I}^{\mathrm{cf}}_λ/σ$, where $σ…
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We show that every monoid which generates a variety containing $L_5^1$ and is contained in the variety generated by $L_\ell^1$ for some $\ell \ge 5$ is non-finitely based. We establish this result by analyzing $τ$-terms for $M$ where $τ$ is certain non-trivial congruence on the free semigroup, that is, we…
Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify these monoids up to isomorphism, determine their idempotents and show that they are $\mathcal{J}$-trivial. For type $A$ we show that Catalan numbers appear as the maximal cardinality of our monoids, in which case the corresponding monoid is isomorphic to the monoid of all order-preserving and order-decreasing total transformations on a finite chain. We construct various representations of these monoi…
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.
Let X be a set and Bin(X) the set of all binary operations on X. We say that a subset of Bin(X) is a distributive set of operations if all pairs of elements are right distributive. J.Przytycki posed the question of which groups can be realized as distributive sets. The initial guess that any group may be embedded into Bin(X) for some X was complicated by an observation that if a binary operation is idempotent (a*a=a), then it commutes with every element of S. The first noncommutative subgroup of Bin(X), the symmetric group on three elements, was found in October of 2011 by Y.Berman. We show th…
Let $\mathcal P_X$ and $\mathcal S_X$ be the partition monoid and symmetric group on an infinite set $X$. We show that $\mathcal P_X$ may be generated by $\mathcal S_X$ together with two (but no fewer) additional partitions, and we classify the pairs $α,β\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal S_X\cup\{α,β\}$. We also show that $\mathcal P_X$ may be generated by the set $\mathcal E_X$ of all idempotent partitions together with two (but no fewer) additional partitions. In fact, $\mathcal P_X$ is generated by $\mathcal E_X\cup\{α,β\}$ if and only if it is generated by…
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 …
In this paper we study the structure of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ of cofinite partial isometries of the $n$-th power of the set of positive integers $\mathbb{N}$ with the usual metric for a positive integer $n\geqslant 2$. We describe the elements of the monoid $\mathbf{I}\mathbb{N}_{\infty}^n$ as partial transformation of $\mathbb{N}^n$, the group of units and the subset of idempotents of the semigroup $\mathbf{I}\mathbb{N}_{\infty}^n$, the natural partial order and Green's relations on $\mathbf{I}\mathbb{N}_{\infty}^n$. In particular we show that the quotient semigroup $\m…
Let $\mathbb{N}^{2}_{\leqslant}$ be the set $\mathbb{N}^{2}$ with the partial order defined as the product of usual order $\leq$ on the set of positive integers $\mathbb{N}$. We study the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ of monotone injective partial selfmaps of $\mathbb{N}^{2}_{\leqslant}$ having cofinite domain and image. We describe properties of elements of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^2_{\leqslant})$ as monotone partial bijections of $\mathbb{N}^{2}_{\leqslant}$ and show that the group of units of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^…
Let $n$ be a positive integer $\geqslant 2$ and $\mathbb{N}^n_{\leqslant}$ be the $n$-th power of positive integers with the product order of the usual order on $\mathbb{N}$. In the paper we study the semigroup of injective partial monotone selfmaps of $\mathbb{N}^n_{\leqslant}$ with cofinite domains and images. We show that the group of units $H(\mathbb{I})$ of the semigroup $\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$ is isomorphic to the group $\mathscr{S}_n$ of permutations of an $n$-element set, and describe the subsemigroup of idempotents of $\mathscr{P\!O}\!_{\infty}(\mathbb{N}…
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Department of Mathematics and Statistics, Chongqing Technology and Business University（重庆工商大学数学与统计学院）；Institute of Natural Sciences and Mathematics, Ural Federal University（乌拉尔联邦大学自然科学与数学研究所）
