Semigroup Closures of Finite Rank Symmetric Inverse Semigroups
Oleg Gutik, Jimmie Lawson, Dušan Repovš
备注：
With the participation of the new coauthor - Jimmie Lawson - the manuscript has been substantially revised and expanded. Accordingly, we have also changed the manuscript title
We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations $\mathscr{I}_λ^n$ of the rank $\leqslant n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.
We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group $G$ is $H$-closed in the class of semitopological inverse semigroups with continuous inversion if and only if $G$ is compact, a Hausdorff linearly ordered topological semilattice $E$ is $H$-closed in the class of semitopological semilattices if and only if $E$ is $H$-closed in the class of topological semilattices, and a topological Brandt $λ^0$-extension of $S$ is (absolutely) $H$-closed i…
In the paper we study pseudocompact primitive topological inverse semigroups. We describe the structure of pseudocompact primitive topological inverse semigroups and show that a Tychonoff product of a family of pseudocompact primitive topological inverse semigroups is a pseudocompact topological space. Also we prove that the Stone-Čech compactification of a pseudocompact primitive topological inverse semigroup is a compact primitive topological inverse semigroup.
We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.
In this paper we show that polycyclic monoids are universal objects in the class of graph inverse semigroups. In particular, we prove that a graph inverse semigroup $G(E)$ over a directed graph $E$ embeds into the polycyclic monoid $\mathcal{P}_λ$ where $λ=|G(E)|$. We show that each graph inverse semigroup $G(E)$ admits the coarsest inverse semigroup topology $τ$. Moreover, each injective homomorphism from $(G(E),τ)$ to the $(\mathcal{P}_{|G(E)|},τ)$ is a topological embedding.
We find necessary and sufficient conditions on an (inverse) semigroup $X$ under which its semigroups of maximal linked systems $λ(X)$, filters $ϕ(X)$, linked upfamilies $N_2(X)$, and upfamilies $\upsilon(X)$ are inverse.
We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions when the maximal subgroup of an inverse primitive feebly compact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt $λ^0$-extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular we show that inversion in a quasi-regular primitive inverse feebly com…
Given a regular language L, we effectively construct a unary semigroup that recognizes the topological closure of L in the free unary semigroup relative to the variety of unary semigroups generated by the pseudovariety R of all finite R-trivial semigroups. In particular, we obtain a new effective solution of the separation problem of regular languages by R-languages.
In the paper we study the semigroup $\mathscr{C}_{\mathbb{Z}}$ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup $\mathscr{C}_{\mathbb{Z}}$ and prove that every non-trivial congruence $\mathfrak{C}$ on the semigroup $\mathscr{C}_{\mathbb{Z}}$ is a group congruence, and moreover the quotient semigroup $\mathscr{C}_{\mathbb{Z}}/\mathfrak{C}$ is isomorphic to a cyclic group. Also we show that the semigroup $\mathscr{C}_{\mathbb{Z}}$ as a Hausdorff semitopological semigroup admits only the discrete topology. Next we study the closure $\oper…
We study the semigroup extension $\mathscr{I}_λ^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_λ^n(S)$ and $\overline{\mathscr{I}_λ^n}(S)$ show that the semigroup $\mathscr{I}_λ^n(S)$ ($\overline{\mathscr{I}_λ^n}(S)$) is regular, orthodox, inverse or stable if and only if so is $S$. Green's relations are described on the semigroup $\mathscr{I}_λ^n(S)$ for an arbitrary monoid $S$. We introduce the conception of a semigroup with strongly tight ideal series, and proved that for any infi…
In this paper we investigate locally compact semitopological graph inverse semigroups. Our main result is the following: if a directed graph $E$ is strongly connected and contains a finite amount of vertices then a locally compact semitopological graph inverse semigroup $G(E)$ is either compact or discrete. This result generalizes results of Gutik and Bardyla who proved the above dichotomy for locally compact semitopological polycyclic monoids $\mathcal{P}_1$ and $\mathcal{P}_λ$, respectively.
We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy for $T_1$ locally compact semitopological …
We give a constructive treatment of some basic concepts and results in semigroup theory. Focusing on semigroups equipped with an apartness relation, we give analogues, from the point of view of apartness, of several classical constructions and results, including transitive closure and congruence closure, free semigroups, periodicity, Rees factors, and Green's relations.
As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $\mathscr{I}_{X}$, a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse algebras is developed. This class of inverse algebras includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.
A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element $a\in S$ are $\mathcal{R}$-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.
The purpose of this paper is to study the generalization of inverse semigroups (without order). An ordered semigroup S is called an inverse ordered semigroup if for every a 2 S, any two inverses of a are H-related. We prove that an ordered semigroup is complete semilattice of t-simple ordered semigroups if and only if it is completely regular and inverse. Furthermore characterizations of inverse ordered semigroups have been characterized by their ordered idempotents.
We prove that if $S$ is an $E$-solid locally inverse semigroup, and $ρ$ is an inverse semigroup congruence on $S$ such that the idempotent classes of $ρ$ are completely simple semigroups then $S$ is embeddable into a $λ$-semidirect product of a completely simple semigroup by $S/ρ$. Consequently, the $E$-solid locally inverse semigroups turn out to be, up to isomorphism, the regular subsemigroups of $λ$-semidirect products of completely simple semigroups by inverse semigroups.
This paper studies the class of automaton semigroups from two perspectives: closure under constructions, and examples of semigroups that are not automaton semigroups. We prove that (semigroup) free products of finite semigroups always arise as automaton semigroups, and that the class of automaton monoids is closed under forming wreath products with finite monoids. We also consider closure under certain kinds of Rees matrix constructions, strong semilattices, and small extensions. Finally, we prove that no subsemigroup of $(\mathbb{N}, +)$ arises as an automaton semigroup. (Previously, $(\mathb…
We define and study the notion of a crossed module over an inverse semigroup and the corresponding $4$-term exact sequences, called crossed module extensions. For a crossed module $A$ over an $F$-inverse monoid $T$, we show that equivalence classes of admissible crossed module extensions of $A$ by $T$ are in a one-to-one correspondence with the elements of the cohomology group $H^3_\le(T^1,A^1)$.
In this paper we discuss graph inverse semigroups which are constucted from a directed graphs and study several interesting properties of graph inverse semigroups such as the nature of its idempotents, the structure of semilattice of idempotents and the like. A necessary and sufficient condition on directed graphs to have the corresponding graph inverse semigroups are primitive inverse semigroups is provided. Further the general form of elements in local submonoids of the graph inverse semigroups and the Green s relations in graph inverse semigroups in graph theoretic terms are also described
We study two classes of inverse semigroups built from directed graphs, namely graph inverse semigroups and a new class of semigroups that we refer to as Leavitt inverse semigroups. These semigroups are closely related to graph $C^*$-algebras and Leavitt path algebras. We provide a topological characterization of the universal groups of the local submonoids of these inverse semigroups. We study the relationship between the graph inverse semigroups of two graphs when there is a directed immersion between the graphs. We describe the structure of graphs that admit a directed cover or directed imme…
In the paper we study (countably) compact and (absolutely) $H$-closed primitive topological inverse semigroups. We describe the structure of compact and countably compact primitive topological inverse semigroups and show that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup.
As part of his study of representations of the polycylic monoids, M.V. Lawson described all the closed inverse submonoids of a polycyclic monoid $P_n$ and classified them up to conjugacy. We show that Lawson's description can be extended to closed inverse subsemigroups of graph inverse semigroups. We then apply B. Schein's theory of cosets in inverse semigroups to the closed inverse subsemigroups of graph inverse semigroups: we give necessary and sufficient conditions for a closed inverse subsemigroup of a graph inverse semigroup to have finite index, and determine the value of the index when …
In a group $G$, elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $g^{-1} ag=b$. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$, which we will write as $a\sim_{\mathrm{i}} b$, if there exists $g\in S^1$ such that $g^{-1} ag=b$ and $gbg^{-1} =a$. The purpose of this paper is to study the conjugacy $\sim_{\mathrm{i}}$ in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAlli…
Inverse graph semigroups were defined by Ash and Hall in 1975. They found necessary and sufficient conditions for the semigroups to be congruence free. In this paper we give a description of congruences on a graph inverse semigroup in terms of the underlying graph. As a consequence, we show that the inverse graph semigroup of a finite graph is congruence Noetherian.
We show that all of the Schützenberger complexes of an Adian inverse semigroup are finite if the Schützenberger complex of every positive word is finite. This enables us to solve the word problem for certain classes of Adian inverse semigroups (and hence for the corresponding Adian semigroups and Adian groups).
Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the lattice of congruences on a graph inverse semigroup is upper semimodular but not lower semimodular.
We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a finite cycle. Finally, by way of application, we prove that finite full transformation semigroups have no cover by inverse semigroups that is closed under intersection.
While every group is isomorphic to a transitive group of permutations, the analogous property fails for inverse semigroups: not all inverse semigroups are isomorphic to transitive inverse semigroups of one-to-one partial transformations of a set. We describe those inverse semigroups that are.
We study topological properties of the symmetric inverse topological semigroup of finite transformations $\mathscr{I}_λ^n$ of the rank $\leqslant n$. We show that the topological inverse semigroup $\mathscr{I}_λ^n$ is algebraically $h$-closed in the class of topological inverse semigroups. Also we prove that a topological semigroup $S$ with countably compact square $S\times S$ does not contain the semigroup $\mathscr{I}_λ^n$ for infinite cardinal $λ$ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations $\mathscr{I}_λ^n$ of the…
This paper introduces a notion of presentation for locally inverse semigroups and develops a graph structure to describe the elements of locally inverse semigroups given by these presentations. These graphs will have a role similar to the role that Cayley graphs have for group presentations or that Schützenberger graphs have for inverse monoid presentations. However, our graphs have considerable differences with the latter two, even though locally inverse semigroups generalize both groups and inverse semigroups. For example, the graphs introduced here are not `inverse word graphs'. Instead, th…
An algebra has the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.
We give a simple construction involving partial actions which permits us to obtain an easy proof of a weakened version of L. O'Carroll's theorem on idempotent pure extensions of inverse semigroups.
Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterise the categories that correspond to locally inverse semigroups in the realm of Nambooripad's cross-connection theory. Further, we specialise our cross-connection description of locally inverse semigroups to inverse semigroups and completely 0-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann-Schein-Nambooripad Theorem; for com…
A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form.
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.
We study the path category of an inverse semigroup admitting unique maximal idempotents and give an abstract characterization of the inverse semigroups arising from zigzag maps on a left cancellative category. As applications we show that every inverse semigroup is Morita equivalent to an inverse semigroup of zigzag maps and hence the class of Cuntz-Krieger $C^*$-algebras of singly aligned categories include the tight $C^*$-algebras of all countable inverse semigroups, up to Morita equivalence.
Inverses semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green's relation H, or in terms of the set of group invertible elements H(S), that allows a study of non-regular semigroups. We then study the interplays between these new classes of semigroups, as well as with known classes of semigroups (notably inverse, orthodox and cryptic semigroups)
This is an overview of the basics of inverse semigroup theory written for the Workshop on Semigroups and Categories held at the University of Ottawa in 2010.
For any graph inverse semigroup $G(E)$ we describe subsemigroups $D^0=D\cup\{0\}$ and $J^0=J\cup\{0\}$ of $G(E)$ where $D$ and $J$ are arbitrary $\mathcal{D}$-class and $\mathcal{J}$-class of $G(E)$, respectively. In particular, we prove that for each $\mathcal{D}$-class $D$ of a graph inverse semigroup over an acyclic graph the semigroup $D^0$ is isomorphic to a semigroup of matrix units. Also we show that for any elements $a,b$ of a graph inverse semigroup $G(E)$, $J_a\cdot J_b\cup J_b\cdot J_a\subset J_b^0$ if there exists a path $w$ such that $s(w)\in J_a$ and $r(w)\in J_b$.
In 1978, Munn proved that a bounded complex representation of an inverse semigroup is semiunitary and completely reducible. We consider the converse question in the finite case. We provide some sufficient conditions from a finite regular semigroup to a finite inverse semigroup.
Let Γ be a directed graph and Inv(Γ) be the graph inverse semigroup of Γ. Luo and Wang [7] showed that the congruence lattice C(Inv(Γ)) of any graph inverse semigroup Inv(Γ) is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell characterized the directed graph Γ for which C(Inv(Γ)) is lower semimodular [2]. In the present paper, we show that the lower semimodularity, modularity and distributivity in the congruence lattice C(Inv(Γ)) of any graph inverse semigroup Inv(Γ) are equivalent.
The idempotent problem of a finitely generated inverse semigroup is the formal language of all words over the generators representing idempotent elements. This note proves that a finitely generated inverse semigroup with regular idempotent problem is necessarily finite. This answers a question of Gilbert and Noonan Heale, and establishes a generalisation to inverse semigroups of Anisimov's Theorem for groups.
Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. We explore their connections with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions).
For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept to consider quotients of inverse semigroup. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quo…
A celebrated result of J. Thompson says that if a finite group $G$ has a fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.
This paper enriches the list of known properties of congruence sequences starting from the universal relation and successively performing the operators lower $k$ and lower $t$. Two series of inverse semigroups, namely $\ker{α_n}$-is-Clifford semigroups and $β_n$-is-over-$E$-unitary semigroups, are investigated. Two congruences, namely $α_{n+2}$ and $β_{n+2}$, are found to be the least $\ker{α_n}$-is-Clifford and least $β_n$-is-over-$E$-unitary congruences on $S$, respectively. A new system of implications is established for the quasivarieties of inverse semigroups induced by the min network.
