Search

For a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, 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 inverse semigroup with covering number $n$, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well.

For a semigroup S, the covering number of S with respect to semigroups, σs (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 ≥ 2, there exists an inverse semigroup with covering number n, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well. [ABSTRACT FROM AUTHOR]

Let 𝒞ℕ be a monoid which is generated by the partial shift α : n↦n +1 of the set of positive integers ℕ and its inverse partial shift β : n + 1 ↦n. In this paper we prove that if S is a submonoid of the monoid Iℕ∞ of all partial cofinite isometries of positive integers which contains Cscr;ℕ as a submonoid then every Hausdorff locally compact shift-continuous topology on S with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup S with an adjoined compact ideal.

We give efficient conditions under which a C*-subalgebra A ⊆ B separates ideals in a C*-algebra B, and B is purely infinite if every positive element in A is properly infinite in B. We specialise to the case when B is a crossed product for an inverse semigroup action by Hilbert bimodules or a section C*-algebra of a Fell bundle over an étale, possibly non-Hausdorff, groupoid. Then our theory works provided B is the recently introduced essential crossed product and the action is essentially exact and residually aperiodic or residually topologically free. These last notions are developed in the article. [ABSTRACT FROM AUTHOR]

We study the semigroup BFnω, which is introduced in the paper [Visnyk Lviv Univ. Ser. Mech.- Mat. 2020, 90, 5–19 (in Ukrainian)], in the case when the ω-closed family Fn generated by the set {0, 1, . . ., n}. We show that the Green relations D and J coincide in BFnω, the semigroup BFnω is isomorphic to the semigroup In+1 ω (conv −−→) of partial convex order isomorphisms of (ω, 6) of the rank 6 n + 1, and BFnω admits only Rees congruences. Also, we study shift-continuous topologies on the semigroup BFnω . In particular, we prove that for any shift-continuous T1 -topology τ on the semigroup BFnω every non-zero element of BFnω is an isolated point of (BFnω, τ), BFnω admits the unique compact shift-continuous T1 -topology, and every ωd-compact shift-continuous T1 -topology is compact. We describe the closure of the semigroup BFnω in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup BFnω is H-closed in the class of Hausdorff topological semigroups. [ABSTRACT FROM AUTHOR]

Let SS be an inverse semigroup with the set of idempotents EE. In this article, we find necessary and sufficient conditions for the weighted semigroup algebra l1(S,ω){l}^{1}\left(S,\omega ) to be module approximately amenable (contractible) and module character amenable (as l1(E){l}^{1}\left(E)-module).

Let BFω be the bicyclic semigroup extension for the family F of ω-closed subsets of ω which is introduced in [19]. We study topologizations of the semigroup BFω for the family F of inductive ω-closed subsets of ω. We generalize Eberhart-Selden and Bertman-West results about topologizations of the bicyclic semigroup [6,12] and show that every Hausdorff shift-continuous topology on the semigroup BFω is discrete and if a Hausdorff semitopological semigroup S contains BFω as a proper dense subsemigroup then S\BFω is an ideal of S. Also, we prove the following dichotomy: every Hausdorff locally compact shift-continuous topology on BFω with an adjoined zero is either compact or discrete. As a consequence of the last result we obtain that every Hausdorff locally compact semigroup topology on BFω with an adjoined zero is discrete and every Hausdorff locally compact shift-continuous topology on the semigroup BFω⊔ I with an adjoined compact ideal I is either compact or the ideal I is open, which extends many results about locally compact topologizations of some classes of semigroups onto extensions of the semigroup BFω. [ABSTRACT FROM AUTHOR]

We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents. [ABSTRACT FROM AUTHOR]

Let S be a commutative (not necessary unital) inverse semigroup with the set of idempotents E then ℓ¹(S) is a commutative Banach ℓ¹(E)-module with canonical actions . Recently, it is shown that the triangular Banach algebra T = [ ℓ¹(S) M ℓ¹(S) ] is (n)-weakly ℓ¹(E)-module amenable, provided that M = ℓ¹(S) and S is unital or E satisfies condition Dk for some K ∈ ℕ . In this paper, we show that T is (2n + 1)- weakly ℓ¹(E)-module amenable, without any additional conditions on S and E, if M is a certain quotient space of ℓ¹(S). [ABSTRACT FROM AUTHOR]

In this short note, we give some counter examples which show that [11, Proposition 3.5] is not true. As a consequence, the arguments in [11, Proposition 4.10] is not valid.

In this paper we study submonoids of the monoid $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ of almost monotone injective co-finite partial selfmaps of positive integers $\mathbb{N}$. Let $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ be a submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which consists of cofinite monotone partial bijections of $\mathbb{N}$ and $\mathscr{C}_{\mathbb{N}}$ be a subsemigroup of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which is generated by the partial shift $n\mapsto n+1$ and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of $\mathscr{I}_{\infty}^{\!\nearrow}(\mathbb{N})$ which contains the semigroup $\mathscr{C}_{\mathbb{N}}$ is the identity map. We construct a submonoid $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ with the following property: if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathbf{I}\mathbb{N}_{\infty}^{[\underline{1}]}$ as a submonoid, then every non-identity congruence $\mathfrak{C}$ on $S$ is a group congruence. We show that if $S$ is an inverse submonoid of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ such that $S$ contains $\mathscr{C}_{\mathbb{N}}$ as a submonoid then $S$ is simple and the quotient semigroup $S/\mathfrak{C}_{\mathbf{mg}}$, where $\mathfrak{C}_{\mathbf{mg}}$ is the minimum group congruence on $S$, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of $\mathscr{I}_{\infty}^{\,\Rsh\!\!\nearrow}(\mathbb{N})$ which contain $\mathscr{C}_{\mathbb{N}}$ and embeddings of such semigroups into compact-like topological semigroups.

Higson proved that every homotopy invariant, stable and split exact functor from the category of C*-algebras to an additive category factors through Kasparov's KK-theory. By adapting a group equivariant generalization of this result by Thomsen, we generalize Higson's result to the inverse semigroup and locally compact, not necessarily Hausdorff groupoid equivariant setting. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of the monoid Iℕn ∞ of cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n ě 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕ

In this paper we study submonoids of the monoid j↗∞(N) of almost monotone injective cofinite partial selfmaps of positive integers N. Let j↗∞(N) be a submonoid of j↗∞(N) which consists of cofinite monotone partial bijections of N and b N be a subsemigroup of j↗∞(N) which is generated by the partial shift n 7→ n + 1 and its inverse partial map. We show that every automorphism of a full inverse subsemigroup of j↗∞(N) which contains the semigroup bN is the identity map. We construct a submonoid IN [1] I ∞ (N) with the following property: if S is an inverse submonoid of j↗∞(N) such that S contains IN [1] ∞ as a submonoid, then every non-identity congruence C on S is a group congruence. We show that if S is an inverse submonoid of j↗∞(N) such that S contains bN as a submonoid then S is simple and the quotient semigroup S/Cmg, where Cmg is the minimum group congruence on S, is isomorphic to the additive group of integers. Also, we study topologizations of inverse submonoids of j↗∞(N) which contain bN and embeddings of such semigroups into compact-like topological semigroups. [ABSTRACT FROM AUTHOR]

We give sufficient conditions when a topological inverse $\lambda$-polycyclic monoid $P_{\lambda}$ is absolutely $H$-closed in the class of topological inverse semigroups. For every infinite cardinal $\lambda$ we construct the coarsest semigroup inverse topology $\tau_{mi}$ on $P_\lambda$ and give an example of a topological inverse monoid $S$ which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.

In recent years, lots of papers have been published on module amenability. In this paper, our main aim is to study the homological properties of various module derivations and prove some results about module amenability. So this paper continous a line investigation in [3], [4] for Banach algebras.

The class of С*-algebras generated by the elementary inverse semigroup and being deformations of the Toeplitz algebra has been introduced and studied. The properties of these algebras have been investigated. All their irreducible representations and automorphism groups have been described. These algebras have been proved to be Z-graded С*-algebras. For a certain class of algebras in the family under consideration the compact quantum semigroup structure has been constructed.

Abstract We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend to étale groupoids, obtaining a duality with an elementary class of inverse semigroups. [ABSTRACT FROM AUTHOR]

The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors - a natural generalization of inverse semigroups and semigroup homomorphisms - and extend the ESN Theorem to an equivalence between this category and the category of locally complete inductive groupoids and locally inductive functors. From the proof of this extension, we also generalize the ESN Theorem to an equivalence between the category of inverse semicategories and the category of locally inductive groupoids and to an equivalence between the category of inverse categories with oplax functors and the category of locally complete inductive groupoids and ordered functors. [ABSTRACT FROM AUTHOR]

In view of the Vagner-Preston theorem, the inverse semigroups can be naturally viewed as semigroups of partial monomorphisms of a given set X. The paper provides an algorithm to describe all indecomposable projective representations of any semigroup of partial monomorphisms of a finite set X that are compatible with a given relation on X contained in a total order on X. The algorithm yields an explicit description of the structure of regular representations of such a given semigroup by exhibiting a complete set of primitive orthogonal idempotents of the respective semigroup algebra and thus presenting the quivers of the respective semigroup algebra. It is at this stage that one can consider questions related to the representation type of a given semigroup and provide in some cases a complete classification of all indecomposable representations. The methods, including those of the graph semigroups, are prepared to describe representation of sets that are unions of finite chains and diamonds. [ABSTRACT FROM AUTHOR]

For a given integer n=p1α1p2α2⋯pkαk$n = p_1^{\alpha _1 } p_2^{\alpha _2 } \cdots p_k^{\alpha _k }$ (k ≥ 2), we give here a class of finitely presented finite monoids of order n. Indeed the monoids Mon(π), where π=〈a1,a2,…,ak|aipiαi=ai, (i=1,2,…,k),aiai+1=ai, (i=1,2,…,k−1)〉.$$\pi = {\langle {a_1 ,a_2 , \ldots ,a_k |a_i^{p_i^{\alpha _i } } = {a_i}, {\left({i = 1,2, \ldots ,k} \right)}, a_i a_{i + 1} = {a_i}, \left({i = 1,2, \ldots ,k - 1} \right)} \rangle} .$$ As a result of this study we are able to classify a wide family of the k-generated p-monoids (finite monoids of order a power of a prime p). An interesting di erence between the center of finite p-groups and the center of finite p-monoids has been achieved as well. All of these monoids are regular and non-commutative.

For self-similar graph actions, we show that isomorphic inverse semigroups associated to a self-similar graph action are a complete invariant for the continuous orbit equivalence of inverse semigroup actions on infinite path spaces.

Directed graphs have long been used to gain an understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Frucht’s Theorem. We investigate two inverse semigroups defined over undirected graphs constructed from the notions of subgraph and vertex set induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these inverse semigroups. We prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism allowing for an algebraic restatement of the Edge Reconstruction Conjecture. [ABSTRACT FROM AUTHOR]

In this note we prove a selection of commutativity theorems for various classes of semigroups. For instance, if in a separative or completely regular semigroup S we have xpyp=ypxp and xqyq=yqxq for all x,y∈S where pp and q are relatively prime, then S is commutative. In a separative or inverse semigroup S, if there exist three consecutive integers i such that (xy)i=xiyi for all x,y∈S, then S is commutative. Finally, if S is a separative or inverse semigroup satisfying (xy)3=x3y3 for all x,y∈S, and if the cubing map x↦x3 is injective, then S is commutative. [ABSTRACT FROM AUTHOR]

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by étale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We prove several main results. To start with, we establish a duality between the category of complete restriction monoids and the category of étale 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 étale 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. A wider class of quantales, called multiplicative Ehresmann quantal frames, is put into a correspondence with those localic categories where the multiplication structure map is semiopen, and all the other structure maps are open. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between étale localic categories and étale 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 étale 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. [ABSTRACT FROM AUTHOR]

In this paper we study and obtain the existence of pseudo-almost automorphic solutions to some classes of second-order abstract differential equations on a Hilbert space. To illustrate our abstract results, we discuss the existence of pseudo almost automorphic solutions to the N-dimensional Sine-Gordon boundary value problem.En este trabajo se estudia y obtiene la existencia de soluciones casi-seudo automorfas a algunas clases de ecuaciones diferenciales abstractas de segundo orden en un espacio de Hilbert. Para ilustrar nuestros resultados abstractos, se discute la existencia de soluciones casi-seudo automorfas en el problema de contorno N-dimensional de Sine-Gordon.

We study the module amenability of Banach modules. This is a natural generalization of Johnson’s amenability of Banach algebras. As an example we show that for a discrete abelian group G, l p(G) is amenable as an l¹(G)-module if and only if G is amenable, where l¹(G) is a Banach algebra with pointwise multiplication.Se estudia el módulo de receptividad de los módulos de Banach. Esta es una generalización natural de la receptividad de Johnson de las álgebras de Banach. Como ejemplo se muestra que para un grupo abeliano discreto G l p(G) es receptivo como un G l p(G)- módulo, si y sólo si G es receptivo, donde l¹(G) es un álgebra de Banach con producto punto.

A corpid is a ring (K, +, ·), different from zero, such that (K, ·) is an inverse semigroup. We define an order relation and the notion of simple element. By this we prove several results and a characterization of corpids.

A corpid is a ring (K, +, ·), different from zero, such that (K, ·) is an inverse semigroup. We prove a characterization of corpids, some remarkable results about the lattice of the idempotents and other results concerning the idempotents and the zero divisors of a corpid. Finally, we are studiyng a notable set of ideals of corpids through which we study the endomorfisms of a corpid.

We study a remarkable class of rings, which we call corpids, that is the rings, different from zero, (K;+; .); such that (K; .) is an inverse semi-group (or groupid, which is the name given by G. Tallini [4]). The inverse semigroup has been defined and called generalized group, indipendently by Viktor Vladimirovich Vagner [6] in the Soviet Union and by Gordon Preston in the Great Britain [3].

Given a graph E , an action of a group G on E , and a G -valued cocycle φ on the edges of E , we define a C*-algebra denoted O G , E , which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup S G , E built naturally from the triple ( G , E , φ ) . As a tight C*-algebra, O G , E is also isomorphic to the full C*-algebra of a naturally occurring groupoid G tight ( S G , E ) . We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which O G , E is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our O G , E , and many of their known properties are shown to follow from our general theory. [ABSTRACT FROM AUTHOR]

We prove a version of Wordingham's theorem for left regular representations in the setting of Fell bundles over inverse semigroups and use this result to discuss the various associated cross sectional C*-algebras. [ABSTRACT FROM AUTHOR]

We give sufficient conditions when a topological inverse λ-polycyclic monoid Pλ is absolutely Hclosed in the class of topological inverse semigroups. For every infinite cardinal λ we construct the coarsest semigroup inverse topology 훕mi on Pλ and give an example of a topological inverse monoid S which contains the polycyclic monoid P2 as a dense discrete subsemigroup. [ABSTRACT FROM AUTHOR]