Search

Motivated by recent interest to $F$-inverse monoids, on the one hand, and to restriction and birestriction monoids, on the other hand, we initiate the study of $F$-birestriction monoids as algebraic structures in the enriched signature $(\cdot, \, ^*, \,^+, \,m,1)$ where the unary operation $m$ maps each element to the maximum element of its $\sigma$-class. We find a presentation of the free $F$-birestriction monoid ${\mathsf{FFBR}}(X)$ as a birestriction monoid ${\mathcal F}$ over the extended set of generators $X\cup\overline{X^+}$ where $\overline{X^+}$ is a set in a bijection with the free semigroup $X^+$ and encodes the maximum elements of (non-projection) $\sigma$-classes. This enables us to show that ${\mathsf{FFBR}}(X)$ decomposes as the partial action product \mbox{$E({\mathcal I})\rtimes X^*$} of the idempotent semilattice of the universal inverse monoid ${\mathcal I}$ of ${\mathcal F}$ partially acted upon by the free monoid $X^*$. Invoking Sch\"utzenberger graphs, we prove that the word problem for ${\mathsf{FFBR}}(X)$ and its strong and perfect analogues is decidable. Furthermore, we show that ${\mathsf{FFBR}}(X)$ does not admit a geometric model based on a quotient of the Margolis-Meakin expansion $M({\mathsf{FG}}(X), X\cup \overline{X^+})$ over the free group ${\mathsf{FG}}(X)$, but the free perfect $X$-generated $F$-birestriction monoid admits such a model., Comment: 33 pages

We extend the equivalence by Cockett and Garner between restriction monoids and ample categories to the biunary and non-unital setting of Boolean range semigroups which are non-unital one-object versions of range categories. We show that Boolean range semigroups are equivalent to ample topological categories where the range map $r$ is open, and \'etale Boolean range semigroups are equivalent to biample topological categories. These results yield the equivalence between \'etale Boolean range semigroups and Boolean birestriction semigroups and a characterization when a Boolean restriction semigroup admits a compatible cosupport operation. We also recover the equivalence between Boolean birestriction semigroups and biample topological categories by Kudryavtseva and Lawson. Our technique builds on the usual constructions relating inverse semigroups with ample topological groupoids via germs and slices., Comment: 39 pages

We show that the universal $X$-generated $F$-inverse monoid $F(G)$, where $G$ is an $X$-generated group, introduced by Auinger, Szendrei and the first-named author, arises as a quotient inverse monoid of the Margolis-Meakin expansion $M(G, X\cup \overline{G})$ of $G$, with respect to the extended generating set $X\cup \overline{G}$, where $\overline{G}$ is a bijective copy of $G$ which encodes the $m$-operation in $F(G)$. The construction relies on a certain dual-closure operator on the semilattice of all finite and connected subgraphs containing the origin of the Cayley graph $Cay(G, X\cup {\overline{G}})$ and leads to a new and simpler proof of the universal property of $F(G)$., Comment: 11 pages, final version

This is an expository paper which provides a quick introduction to Boolean inverse semigroups and their type monoids, with the emphasis on techniques and insights of the theory, and also treats the connection of the type monoid ${\mathrm{Typ}}(S)$ of a Boolean inverse semigroup $S$ with the monoid $V(K\langle S\rangle)$ of the ring $K\langle S\rangle$ assigned to $S$. We give original direct and simple proofs of some known results, such as the structure of semisimple Boolean inverse semigroups, the presentation of the type monoid by generalized rook matrices. We also prove that the type monoid of the tight Booleanization of a graph inverse semigroup is isomorphic to the graph monoid of this semigroup.

We propose two universal constructions of globalization of a partial action of a semigroup on a set, satisfying certain conditions which arise in Morita theory of semigroups. One of the constructions is based on the tensor product of a partial semigroup act with the semigroup and generalizes the globalization construction of strong partial actions of monoids due to Megrelishvili and Schr\"oder. It produces the initial object in an appropriate caterory of globalizations of a given partial action. The other construction involves ${\mathrm{Hom}}$-sets and is novel even in the monoid setting. It produces the terminal object in an appropriate category of globalizations. While in the group case the results of the two constructions are isomorphic, they can be far different in the monoid case., Comment: Revised version

We propose a notion of a proper Ehresmann semigroup based on a three-coordinate description of its generating elements governed by certain labelled directed graphs with additional structure. The generating elements are determined by their domain projection, range projection and $\sigma$-class, where $\sigma$ denotes the minimum congruence that identifies all projections. We prove a structure result on proper Ehresmann semigroups and show that every Ehresmann semigroup has a proper cover. Our covering monoid turns out to be isomorphic to that from the work by Branco, Gomes and Gould and provides a new view of the latter. Proper Ehresmann semigroups all of whose elements admit a three-coordinate description are characterized in terms of partial multiactions of monoids on semilattices. As a consequence we recover the two-coordinate structure result on proper restriction semigroups., Comment: 29 pages, revised version

We introduce $X$-to-join representations of inverse semigroups which are a relaxation of the notion of a cover-to-join representation. We construct the universal $X$-to-join Booleanization of an inverse semigroup $S$ as a weakly meet-preserving quotient of the universal Booleanization ${\mathrm B}(S)$ and show that all such quotients of ${\mathrm B}(S)$ arise via $X$-to-join representaions. As an application, we provide groupoid models for the intermediate boundary quotients of the $C^*$-algebra of a Zappa-Sz\'ep product right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker., Comment: 19 pages

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We establish an adjunction between the category ${\mathcal{P}}(S)$ of proper extensions of a restriction semigroup (or, in particular, an inverse semigroup) $S$ and a category ${\mathcal{A}}(S)$ of partial actions of $S$ subject to certain conditions going back to the work of O'Carroll. In the category ${\mathcal{A}}(S)$, we specify two isomorphic subcategories, one being reflective and the other one coreflective, each of which is equivalent to the category ${\mathcal{P}}(S)$., Comment: 32 pages; revised version

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget-Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of respective relatively free inverse monoids. For a monoid $M$, we define ${\mathcal{FR}}_R(M)$ to be the freest two-sided restriction monoid generated by a bijective copy, $M'$, of the underlying set of $M$, such that the inclusion map $\iota\colon M\to {\mathcal{FR}}_R(M)$ is determined by a set of relations, $R$, so that $\iota$ is a premorphism which is weaker than a homomorphism. Our main result states that ${\mathcal{FR}}_R(M)$ can be constructed, by means of a partial action product construction, from $M$ and the idempotent semilattice of ${\mathcal{FI}}_R(M)$, the free $M'$-generated inverse monoid subject to relations $R$. In particular, the semilattice of projections of ${\mathcal{FR}}_R(M)$ is isomorphic to the idempotent semilattice of ${\mathcal{FI}}_R(M)$. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that important properties of ${\mathcal{FR}}_R(M)$ are well agreed with suitable properties of $M$, such as being cancellative or embeddable into a group. We observe that if $M$ is an inverse monoid, then ${\mathcal{FI}}_s(M)$, the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson-Margolis-Steinberg generalized prefix expansion $M^{pr}$. This gives a presentation of $M^{pr}$ and leads to a model for ${\mathcal{FR}}_s(M)$ in terms of the known model for $M^{pr}$., Comment: 27 pages; Proposition 7.9 (pages 24-25) modified, material that follows updated accordingly, remaining material unaffected; minor typos correction

An exponent matrix is an $n\times n$ matrix $A=(a_{ij})$ over ${\mathbb N}^0$ satisfying (1) $a_{ii}=0$ for all $i=1,\ldots, n$ and (2) $a_{ij}+a_{jk}\geq a_{ik}$ for all pairwise distinct $i,j,k\in\{1,\dots, n\}$. In the present paper we study the set ${\mathcal E}_n$ of all non-negative $n\times n$ exponent matrices as an algebra with the operations $\oplus$ of component-wise maximum and $\odot$ of component-wise addition. We provide a basis of the algebra $({\mathcal E}_n, \oplus, \odot,0)$ and give a row and a column decompositions of a matrix $A\in {\mathcal E}_n$ with respect to this basis. This structure result determines all $n\times n$ tiled orders over a fixed discrete valuation ring. We also study automorphisms of ${\mathcal E}_n$ with respect to each of the operations $\oplus$ and $\odot$ and prove that ${\rm Aut}(\mathcal{E}_n,\, \odot ) = {\rm Aut}(\mathcal{E}_n,\, \oplus ) = {\rm Aut}(\mathcal{E}_n,\, \odot ,\oplus ,0) \simeq {\mathcal{S}}_n \times C_2,$$n>2.$, Comment: 17 pages

We show that atoms of the $n$-generated free left-handed skew Boolean intersection algebra are in a bijective correspondence with pointed partitions of non-empty subsets of $\{1,2,\dots, n\}$. Furthermore, under the canonical inclusion into the $k$-generated free algebra, where $k\geq n$, an atom of the $n$-generated free algebra decomposes into an orthogonal join of atoms of the $k$-generated free algebra in an agreement with the containment relation on the respective partitions. As a consequence of these results, we describe the structure of finite free left-handed skew Boolean intersection algebras and express several their combinatorial characteristics in terms of Bell numbers and Stirling numbers of the second kind. We also look at the infinite case. For countably many generators, our constructions lead to the `partition analogue' of the Cantor tree whose boundary is the `partition variant' of the Cantor set.

We study the structure and properties of free skew Boolean algebras. For finite generating sets, these free algebras are finite and we give their representation as a product of primitive algebras and provide formulas for calculating their cardinality. We also characterize atomic elements and central elements and calculate the number of such elements. These results are used to study minimal generating sets of finite skew Boolean algebras. We also prove that the center of the free infinitely generated algebra is trivial and show that all free algebras have intersections., Comment: 24 pages, revised version

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra. For the topos of sets, we show that torsion-free functors on Loganathan's category $L(S)$ of an inverse semigroup $S$ are equivalent to a special class of non-strict representations of $S$, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of $S$. We describe the correspondence between directed and pullback preserving functors on $L(S)$ and transitive and effective representations of $S$, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation of an inverse semigroup in the topos of sheaves ${\mathsf{Sh}}(X)$ on a topological space $X$ as well as outline an approach on how to define such a representation in an arbitrary topos. We prove that the category of filtered functors from $L(S)$ to the topos ${\mathsf{Sh}}(X)$ is equivalent to the category of universal representations of $S$ in ${\mathsf{Sh}}(X)$.

The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods.

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by \'etale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We establish a duality between the category of complete restriction monoids and the category of \'etale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups and \'etale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between \'etale localic categories and \'etale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectral \'etale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work by Lawson and Lenz and by Resende., Comment: 69 pages

We study classes of proper restriction semigroups determined by properties of partial actions underlying them. These properties include strongness, antistrongness, being defined by a homomorphism, being an action etc. Of particular interest is the class determined by homomorphisms, primarily because we observe that its elements, while being close to semidirect products, serve as mediators between general restriction semigroups and semidirect products or $W$-products in an embedding-covering construction. It is remarkable that this class does not have an adequate analogue if specialized to inverse semigroups. $F$-restriction monoids of this class, called ultra $F$-restriction monoids, are determined by homomorphisms from a monoid $T$ to the Munn monoid of a semilattice $Y$. We show that these are precisely the monoids $Y*_mT$ considered by Fountain, Gomes and Gould. We obtain a McAlister-type presentation for the class given by strong dual prehomomorphisms and apply it to construct an embedding of ultra $F$-restriction monoids, for which the base monoid $T$ is free, into $W$-products of semilattices by monoids. Our approach yields new and simpler proofs of two recent embedding-covering results by Szendrei.

We describe right-hand skew Boolean algebras in terms of a class of presheaves of sets over Boolean algebras called Boolean sets, and prove a duality theorem between Boolean sets and etale spaces over Boolean spaces.

We refine Funk's description of the classifying space of an inverse semigroup by replacing his *-semigroups by right generalized inverse *-semigroups. Our proof uses the idea that presheaves of sets over meet semilattices may be characterized algebraically as right normal bands.

We prove that the structure of right generalized inverse semigroups is determined by free \'etale actions of inverse semigroups. This leads to a tensor product interpretation of Yamada's classical struture theorem for generalized inverse semigroups.

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras. From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and so-called override. Our duality shows that there is a canonical choice for this embedding. Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces., Comment: 20 pages

Motivated by recent appearance of multivalued structures in categorification, tropical geometry and other areas, we study basic properties of abstract multisemigroups. We give many new and old examples and general constructions for multisemigroups. Special attention is paid to simple and nilpotent multisemigroups. We also show that "almost all" randomly chosen multivalued binary operations define multisemigroups., Comment: revised version 31 pages

The aim of the present paper is to extend the dualizing object approach to Stone duality to the non-commutative setting of skew Boolean algebras. This continues the study of non-commutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In particular, we construct a series of dual adjunctions between the categories of Boolean spaces and skew Boolean algebras, unital versions of which are induced by dualizing objects ${0,1,..., n+1}$, $n\geq 0$. We describe Eilenberg-Moore categories of the monads of our adjunctions and construct easily understood non-commutative reflections of skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their `twisted product' functor $\omega$., Comment: 17 pages

We establish two duality theorems which refine the classical Stone duality between generalized Boolean algebras and locally compact Boolean spaces. In the first theorem we prove that the category of left-handed skew Boolean algebras whose morphisms are proper skew Boolean algebra homomorphisms is equivalent to the category of \'{e}tale spaces over locally compact Boolean spaces whose morphisms are \'{e}tale space cohomomorphisms over continuous proper maps. In the second theorem we prove that the category of left-handed skew Boolean $\cap$-algebras whose morphisms are proper skew Boolean $\cap$-algebra homomorphisms is equivalent to the category of \'{e}tale spaces with compact clopen equalizers over locally compact Boolean spaces whose morphisms are injective \'{e}tale space cohomomorphisms over continuous proper maps., Comment: 15 pages; an inaccuracy of v1 is corrected (nonzero homomorphisms of skew Boolean algebras are replaced by proper homomorphisms)

We prove that the minimal cardinality of the semitransitive subsemigroup in the singular part $\IS_n\setminus \S_n$ of the symmetric inverse semigroup $\IS_n$ is $2n-p+1$, where $p$ is the greatest proper divisor of $n$, and classify all semitransitive subsemigroups of this minimal cardinality.

Inspired by the results of [R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand model, preprint math.RT arXiv:0709.3962], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the $q$-rook monoid algebra., Comment: 14 pages

We compare three approaches to the notion of conjugacy for semigroups, the first one via the transitive closure of the $uv\sim vu$ relation, the second one via an action of inverse semigroups on themselves by partial transformations, and the third one via characters of finite-dimensional representations., Comment: 8 pages

We classify minimal transitive subsemigroups of the finitary inverse symmetric semigroup modulo the classification of minimal transitive subgroups of finite symmetric groups; and semitransitive subsemigroups of the finite inverse symmetric semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups., Comment: 16 pages

We obtain Schur-Weyl dualities in which the algebras, acting on both sides, are semigroup algebras of various symmetric inverse semigroups and their deformations., Comment: 14 pages

We show how the categorial approach to inverse monoids can be described as a certain endofunctor (which we call the partialization functor) of some category. In this paper we show that this functor can be used to obtain several recently defined inverse monoids, and use it to define a new object, which we call the inverse braid-permutation monoid. A presentation for this monoid is obtained. Finally, we study some abstract properties of the partialization functor and its iterations. This leads to a categorification of a monoid of all order preserving maps, and series of orthodox generalizations of the symmetric inverse semigroup., Comment: 23 pages

We give a monoid presentation in terms of generators and defining relations for the partial analogue of the finite dual inverse symmetric monoid., Comment: 19 pages, 5 figures

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case when $S$ is a monoid, denote by $G$ the group of units of $S$. Then the relation of $G$-conjugacy is defined by $a\sim_G b \iff a=g^{-1}bg$ for certain $g\in G$. We establish the structure of conjugacy classes for regular epigroups (i.e. semigroups such that some power of each element lies in a subgroup). As a corollary we obtain a criterion of conjugacy in terms of $G$-conjugacy for factorizable inverse epigroups. We show that our general conjugacy criteria easily lead to known and new conjugacy criteria for some specific semigroups, among which are the full transformation semigroup and the full inverse symmetric semigroup over a finite set, the linear analogues of these semigroups and the semigroup of finitary partial automatic transformations over a finite alphabet., Comment: 15 pages, 1 figure

We introduce two generalisations of the full symmetric inverse semigroup ${\mathcal{I}}_X$ and its dual semigroup ${\mathcal{I}^{\ast}}_X$ -- inverse semigroups ${\mathcal{PI}^{\ast}}_X$ and ${\overline{\mathcal{PI}^{\ast}}}_X$. Both of them have the same carrier and contain $\IS_X$. Binary operations on ${\mathcal{PI}^{\ast}}_X$ and ${\overline{\mathcal{PI}^{\ast}}}_X$ are reminiscent of the multiplication in ${\mathcal{I}}_X$. We use a convenient geometric way to realise elements from these two semigroups. This enables us to study efficiently their inner properties and to compare them with the corresponding properties of $\IS_X$ and ${\mathcal{I}^{\ast}}_X$., Comment: 24 pages

We construct a family of representations of an arbitrary variant $S_a$ of a semigroup $S$, induced by a given representation of $S$, and investigate properties of such representations and their kernels., Comment: 12 pages

We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids., Comment: 23 pages, revised version

It is proved that the automorphism group of a semigroup being an inflation of its proper subsemigroup decomposes into a semidirect product of two groups one of which is a direct sum of full symmetric groups., Comment: 6 pages

We study the algebraic properties of the series $\mathrm{K}_n$ of semigroups, which is inspired by \cite{Ki} and has origins in convexity theory. In particular, we describe Green's relations on $\mathrm{K}_n$, prove that there exists a faithful representation of $\mathrm{K}_n$ by $n\times n$ matrices with non-negative integer coefficients (and even explicitly construct such a representation), and prove that $\mathrm{K}_n$ does not admit a faithful representation by matrices of smaller size. We also describe the maximal nilpotent subsemigroups in $\mathrm{K}_n$, all isolated and completely isolated subsemigroups, all automorphisms and anti-automorphisms of $\mathrm{K}_n$. Finally, we explicitly construct all irreducible representations of $\mathrm{K}_n$ over any field and describe primitive idempotents in the semigroup algebra (which we prove is basic)., Comment: 26 pages

We study the structure of nilpotent subsemigroups in the semigroup $M(n,\mathbb{F})$ of all $n\times n$ matrices over a field, $\mathbb{F}$, with respect to the operation of the usual matrix multiplication. We describe the maximal subsemigroups among the nilpotent subsemigroups of a fixed nilpotency degree and classify them up to isomorphism. We also describe isolated and completely isolated subsemigroups and conjugated elements in $M(n,\mathbb{F})$., Comment: A revised version of the preprint: A. Kudryavtseva, V. Mazorchuk, Square matrices as a semigroup, Preprint 2002:18, Uppsala University

We classify all cross-sections of Green's relations $\mathcal{L}$ and $\mathcal{R}$ in the Brauer semigroup. The regular behavior of such cross-sections starts from $n=7$. We show that in the regular case there are essentially two different cross-sections and all others are $\mathcal{S}_n$-conjugated to one of these two. We also classify all cross-sections up to isomorphism., Comment: 21 page, 5 figures