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 semitopological semigroup, and every locally compact Hausdorff semigroup topology on $P_λ$ is discrete. The last statement extends results of the paper [33] obtaining for topological inverse graph semigroups. We describe all feebly compact topologies $τ$ on $P_λ$ such that $\left(P_λ,τ\right)$ is a semitopological semigroup and its Bohr compactification as a topological semigroup. We prove that for every cardinal $λ\geqslant 2$ any continuous homomorphism from a topological semigroup $P_λ$ into an arbitrary countably compact topological semigroup is annihilating and there exists no a Hausdorff feebly compact topological semigroup which contains $P_λ$ as a dense subsemigroup.
We prove that a Hausdorff locally compact semitopological bicyclic semigroup with adjoined zero $\mathscr{C}^0$ is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological bicyclic semigroup with an adjoined compact ideal and construct an example which witnesses that a counterpart of the statements does not hold when $\mathscr{C}^0$ is a Čech-complete metrizable topological inverse semigroup.
Semitopological interassociates $\mathscr{C}_{m,n}$ of the bicyclic semigroup $\mathscr{C}(p,q)$ are studied. In particular, we show that for arbitrary non-negative integers $m$, $n$ and every Hausdorff topology $τ$ on $\mathscr{C}_{m,n}$ such that $\left(\mathscr{C}_{m,n},τ\right)$ is a semitopological semigroup, is discrete. Also, we prove that if an interassociate of the bicyclic monoid $\mathscr{C}_{m,n}$ is a dense subsemigroup of a Hausdorff semitopological semigroup $(S,\cdot)$ and $I=S\setminus\mathscr{C}_{m,n}\neq\varnothing$ then $I$ is a two-sided ideal of the semigroup $S$ and show…
In the paper we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, $ω$-boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) to\-pological Brandt $λ_i^0$-extensions of semitopological monoids with zero. In particular we show that if $\big\{ \big(B^0_{λ_i}(S_i),τ^0_{B(S_i)}\big) \colon i\in\mathscr{I}\big\}$ is a family of Hausdorff pseudocompact to\-pological Brandt $λ_i^0$-extensions of pseudocompact semitopological monoids with zero such…
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.
In the paper we investigate topological properties of a topological Brandt $λ^0$-extension $B^0_λ(S)$ of a semitopological monoid $S$ with zero. In particular we prove that for every Tychonoff pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) semitopological monoid $S$ with zero there exists a unique semiregular pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) extension $B^0_λ(S)$ of $S$ and establish theirs Stone-Čech and Bohr compactifications. We also describe a category whose objects are ingredients in the constructions of pseudocompact (resp., coun…
We give sufficient conditions when a topological inverse $λ$-polycyclic monoid $P_λ$ is absolutely $H$-closed in the class of topological inverse semigroups. Also, for every infinite cardinal $λ$ we construct the coarsest semigroup inverse topology $τ_{mi}$ on $P_λ$ and give an example of a topological inverse monoid S which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.
In this paper we consider McAlister semigroups over arbitrary cardinals and investigate their algebraic and topological properties. We show that the group of automorphisms of a McAlister semigroup $\mathcal{M}_λ$ is isomorphic to the direct product $Sym(λ){\times}\mathbb{Z}_2$, where $Sym(λ)$ is the group of permutations of the cardinal $λ$. This fact correlates with the result of Mashevitzky, Schein and Zhitomirski which states that the group of automorphisms of the free inverse semigroup over a cardinal $λ$ is isomorphic to the wreath product of $Sym(λ)$ and $\mathbb{Z}_2$. Each McAlister se…
We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid $C_{n}$ called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid $P_{n}$. The étale topological groupoid associated with $C_{n}$ under non-commutative Stone duality is the usual groupoid associated with the corresponding Cuntz $C^{\ast}$-algebra. We then show that the group $G_{n,1}$ is also the group of auto…
In the paper we introduce a notion of the Bruck-Reilly $λ$-polycyclic extension of a monoid $S$ with a homomorphism $θ$ which is an analogue of the Bruck-Reilly extension of a monoid $S$. We describe idempotens of the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ and Green's relations on $\left(\mathscr{P}_λ(θ,S),*\right)$. It is proved that $\left(\mathscr{P}_λ(θ,S),*\right)$ is a $0$-simple semigroup for any semigroup $S$. We find necessary and sufficient conditions on a monoid $S$ and a homomorphism $θ$ under which the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ is regular, inverse, $0$-b…
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 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…
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.
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 study the semigroup $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$ of monotone injective partial selfmaps of the set of integers having cofinite domain and image. We show that $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$ is bisimple and all of its non-trivial semigroup homomorphisms are either isomorphisms or group homomorphisms. We also prove that every Baire topology $τ$ on $\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z})$ such that $(\mathscr{I}^{\nearrow}_{\infty}(\mathbb{Z}),τ)$ is a Hausdorff semitopological semigroup is discrete and we construct a non-discrete Hausdorff inverse semigroup…
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…
In the paper we study algebraic properties of the monoid $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ of cofinite partial isometries of the set of positive integers $\mathbb{N}$ with the bounded finite noise $j$. For the monoids $\mathbf{I}\mathbb{N}_{\infty}^{\boldsymbol{g}[j]}$ we prove counterparts of some classical results of Eberhart and Selden describing the closure of the bicyclic semigroup in a locally compact topological inverse semigroup. In particular we show that for any positive integer $j$ every Hausdorff shift-continuous topology $τ$ on $\mathbf{I}\mathbb{N}_{\infty}^{\bo…
We define compact automata and show that every language has a unique minimal compact automaton. We also define recognition of languages by compact left semitopological monoids and construct the analogue of the syntactic monoid in this context. For rational languages this reduces to the usual theory of finite automata and finite monoids.
We study the semigroup $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ of monotone injective partial selfmaps of the set of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ having co-finite domain and image, where $L_n\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic product of $n$-elements chain and the set of integers with the usual order. We show that $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is bisimple and establish its projective congruences. We prove that $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is finitely generated, and for…
We study the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$, which is introduced in [O. Gutik and M. Mykhalenych, \emph{On some generalization of the bicyclic monoid}, Visnyk Lviv. Univ. Ser. Mech.-Mat. \textbf{90} (2020), 5--19], in the case when the family $\mathscr{F}$ of subsets of cardinality $\leqslant 1$ in $ω$. We show that $\boldsymbol{B}_ω^{\mathscr{F}}$ is isomorphic to the subsemigroup $\mathscr{B}_ω^{\Rsh}(\boldsymbol{F}_{\min})$ of the Brandt $ω$-extension of the semilattice $\boldsymbol{F}_{\min}$ and describe all shift-continuous feebly compact $T_1$-topologies on the semigroup $\m…
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.
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. I…
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 …
