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The semigroup B 0 is the only 4-element subsemigroup of the 5-element Brandt semigroup B 2 , up to isomorphism. Being an inverse semigroup, the semigroup B 2 can naturally be considered an additively idempotent semiring and B 0 is its subsemiring. We show that the semiring B 0 has a finite basis of identities. [ABSTRACT FROM AUTHOR]

This article starts by relating formations of inverse semigroups, of congruences on inverse semigroups and of congruences on free inverse semigroups. Then, i-formations of idempotent separating congruences and of languages are defined and their classes are proved to be in bijection, and in bijection with f-formations of inverse semigroups. In the second part, the concepts of i-Fitting class of inverse semigroups, of congruences and of languages are introduced. Bijections between s-fitting classes of inverse semigroups, i-fitting classes of congruences and i-fitting classes of languages are presented. Of special relevance are results on formations and Fitting classes induced by the inverse case on the group case. [ABSTRACT FROM AUTHOR]

We describe injective endomorphisms of the inverse semigroup B ω F introduced in [Visn. Lviv. Univ., Ser. Mech.-Mat., Issue 90, 5 (2020)] for the case where the family F n is generated by the set {0,1,..., n}. In particular, it is shown that the semigroup of injective endomorphisms of the semigroup B ω F is isomorphic to (ω, +). Moreover, we describe the structure of the semigroup E n d B λ of all endomorphisms of the semigroup of λ × λ -matrix units B λ . [ABSTRACT FROM AUTHOR]

Relying on the notions of submodular function and partial metric, we introduce normed inverse semigroups as a generalization of normed groups and sup-semilattices equipped with an upper valuation. We define the property of skew-convexity for a metric on an inverse semigroup, and prove that every norm on a Clifford semigroup gives rise to a right-subinvariant and skew-convex metric; it makes the semigroup into a Hausdorff topological inverse semigroup if the norm is cyclically permutable. Conversely, we show that every Clifford monoid equipped with a right-subinvariant and skew-convex metric admits a norm for which the metric topology and the norm topology coincide. We characterize convergence of nets and show that Cauchy completeness implies conditional monotone completeness with respect to the natural partial order of the inverse semigroup. [ABSTRACT FROM AUTHOR]

We show that the category of X-generated E-unitary inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of G. Analogously, we study F-inverse monoids in the extended signature (· , 1 , - 1 , m) , and show that the category of X-generated F-inverse monoids with greatest group image G is equivalent to the category of G-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of G. As an application, we show that presentations of F-inverse monoids in the extended signature can be studied by tools analogous to Stephen's procedure in inverse monoids, in particular, we introduce the notions of F-Schützenberger graphs and P-expansions. [ABSTRACT FROM AUTHOR]

From any directed graph E one can construct the graph inverse semigroup G (E) , whose elements, roughly speaking, correspond to paths in E. Wang and Luo showed that the congruence lattice L (G (E)) of G (E) is upper-semimodular for every graph E, but can fail to be lower-semimodular for some E. We provide a simple characterization of the graphs E for which L (G (E)) is lower-semimodular. We also describe those E such that L (G (E)) is atomistic, and characterize the minimal generating sets for L (G (E)) when E is finite and simple. [ABSTRACT FROM AUTHOR]

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis. [ABSTRACT FROM AUTHOR]

Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. Some applications to the basic dynamical properties of the action are indicated, covering topics as topological transitivity, limit points and periodic points. We then explore the connections of these recurrence sets with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions). [ABSTRACT FROM AUTHOR]

A class of Möbius monoids leads us to Möbius categories of Q 3 -type via a particular breaking process, where Q 3 is a quiver with three arrows (atoms). In this paper we show that a quasi-commutativity regarding composable atoms uniquely determines (via a certain local congruence) a half-factorial broken Möbius category of Q 3 -type as a quotient category of the path category of Q 3 . Some examples shed light on the development of the topic under discussion. On the other hand, the Möbius breaking process can be extended for inverse semigroups as well. The Leech inverse semigroups of the two broken Möbius categories (the path category of Q 3 and its quotient category) are split inverse semigroups in the sense that both are unions of two proper inverse subsemigroups, that is, both are with covering numbers 2. The connections on the two planes (broken Möbius categories and split inverse semigroups) are made on one hand by a local congruence ϱ + of the path category of Q 3 , and on the other hand by a normal inverse subsemigroup G + namely a gauge inverse subsemigroup. This gauge inverse semigroup is a full inverse subsemigroup of the almost Cartesian product B × 0 B N of the bicyciclic semigroup B and the Brandt semigroup B N . Special properties of this almost Cartesian product are examined by comparison with the well known polycyclic monoid. [ABSTRACT FROM AUTHOR]

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.

The present article establishes two contrasting results: a finite involution semigroup is embeddable in some semigroup of complex matrices with conjugate transposition if and only if it is an inverse semigroup, while every finite involution semigroup is embeddable in some semigroup of binary matrices with the skew transposition across the secondary diagonal. [ABSTRACT FROM AUTHOR]

In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree n for n > 0 , equals the sum of the number of cyclic subgroups of the symmetric groups on 1 to n points. We also prove an analogous statement for monogenic subsemigroups of the finite full transformation monoids, as well as monogenic inverse submonoids and subsemigroups of the finite symmetric inverse monoids. [ABSTRACT FROM AUTHOR]

Let S be a pseudo-unitary homogeneous (graded) inverse semigroup with zero 0, that is, an inverse semigroup with zero, and with a family { S δ } δ ∈ Δ of nonzero subsets of S, called components of S, indexed by a partial groupoid Δ , that is, by a set with a partial binary operation, such that S = ⋃ δ ∈ Δ S δ , and: i) S ξ ∩ S η ⊆ { 0 } for all distinct ξ , η ∈ Δ ; ii) S ξ S η ⊆ S ξ η whenever ξ η is defined; iii) S ξ S η ⊈ { 0 } if and only if the product ξ η is defined; iv) for every idempotent element ϵ ∈ Δ , the subsemigroup S ϵ is with identity 1 ϵ ; v) for every x ∈ S there exist idempotent elements ξ , η ∈ Δ such that 1 ξ x = x = x 1 η ; vi) 1 ξ 1 η = 1 ξ η whenever ξ η ∈ Δ is an idempotent element, where ξ , η are idempotent elements of Δ . Let A be a subset of the union of the subsemigroup components of S, which does not contain 0. By Cay (S ∗ , A) we denote a graph obtained from the Cayley graph Cay (S , A) by removing 0 and its incident edges. We characterize vertex-transitivity of Cay (S ∗ , A) and relate it to the vertex-transitivity of its subgraph whose vertex set is S μ \ { 0 } , where μ is the maximum element of the set of all idempotent elements of Δ , with respect to the natural order. [ABSTRACT FROM AUTHOR]

We introduce maximal ordered groupoids and study some of their properties. Also, we use the Ehresmann–Schein–Nambooripad Theorem, which establishes a one-to-one correspondence between inverse semigroups and a class of ordered groupoids, to prove a Galois correspondence for the case of inverse semigroups acting orthogonally on commutative rings. [ABSTRACT FROM AUTHOR]

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]

For n ∈ N , let [ n ] = { 1 , 2 , ... , n } be an n - element set. As usual, we denote by I n the symmetric inverse semigroup on [n], i.e. the partial one-to-one transformation semigroup on [n] under composition of mappings. The crown (cycle) C n is an n-ordered set with the partial order ≺ on [n], where the only comparabilities are 1 ≺ 2 ≻ 3 ≺ 4 ≻ ⋯ ≺ n ≻ 1 or 1 ≻ 2 ≺ 3 ≻ 4 ≺ ⋯ ≻ n ≺ 1. We say that a transformation α ∈ I n is order-preserving if x ≺ y implies that x α ≺ y α , for all x, y from the domain of α . In this paper, we study the inverse semigroup I C n of all partial automorphisms on a finite crown C n . We consider the elements, determine a generating set of minimal size and calculate the rank of I C n . [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]

A semi-lattice is said to be tree-like when any two of its elements are either orthogonal or comparable. Given an inverse semigroup S whose idempotent semi-lattice is tree-like, and such that all tight filters are ultra-filters, we present a necessary and sufficient condition for S to be contracting which looks closer in spirit to the notion of contracting actions than a condition found by the second named author and E. Pardo. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to describe the closure of a finitely generated subgroup of a finitely generated free group in the proabelian topology. Our approach depends heavily on the description of the abelian kernel of a finite inverse semigroup. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the theory of Rota–Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples (S , + , ∘) where (S , +) and (S , ∘) are Clifford semigroups such that a ∘ (b + c) = a ∘ b − a + a ∘ c and a ∘ a − = − a + a , for all a , b , c ∈ S. To each algebraic structure is associated a set-theoretic solution of the Yang–Baxter equation that has a behaviour near to the bijectivity and non-degeneracy. Drawing from the theory of Clifford semigroups, we provide methods for constructing dual weak braces and deepen some structural aspects, including the notion of ideal. [ABSTRACT FROM AUTHOR]

We review various constructions of wreath products of groups, semigroups, Lie algebras and associative algebras and discuss their realizations in matrix wreath products of associative algebras. As an application we prove a new version of Evans's embedding theorem [T. Evans, Embedding theorems for multiplicative systems and projective geometries, Proc. American Math. Soc. 3 (1952) 614–620]. [ABSTRACT FROM AUTHOR]

We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups. [ABSTRACT FROM AUTHOR]

Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterize the categories that correspond to locally inverse semigroups in the realm of Nambooripad's cross-connection theory. Further, we specialize 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 completely 0 -simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools. [ABSTRACT FROM AUTHOR]

We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as 'slice-sections' of groupoid bundles. [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 are interested in properties, especially injectivity (in the sense of category theory), of the ternary rings of operators generated by certain subsets of an inverse semigroup via the regular representation. We determine all subsets of the extended bicyclic semigroup which are closed under the triple product x y ∗ z (called semiheaps) and show that the weakly closed ternary rings of operators generated by them are injective operator spaces. [ABSTRACT FROM AUTHOR]

We will introduce an inverse semigroup written S A associated with a one-sided topological Markov shift (X A , σ A) for an irreducible matrix A with entries in { 0 , 1 } . We will show that two inverse semigroups S A and S B are isomorphic if and only if the one-sided topological Markov shifts (X A , σ A) and (X B , σ B) are continuous orbit equivalent. As a result, the isomorphism class of the inverse semigroup S A is described in terms of an étale groupoid G A , Cuntz–Krieger algebra O A , and so on. [ABSTRACT FROM AUTHOR]

Let Q be an inverse semigroup. A subsemigroup S of Q is a left I-order in Q and Q is a semigroup of left I-quotients of S if every element in Q can be written as a - 1 b , where a , b ∈ S and a - 1 is the inverse of a in the sense of inverse semigroup theory. If we insist on being able to take a and b to be R -related in Q we say that S is straight in Q and Q is a semigroup of straight left I-quotients of S. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of R and L from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the L -classes of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight. [ABSTRACT FROM AUTHOR]

In this paper, we provide an abstract characterization of the inverse hulls of semigroups associated with Markov shifts. As an application of the characterization we give an example of Markov shifts that are not conjugate, but have isomorphic inverse hulls. [ABSTRACT FROM AUTHOR]

This article explores a generalisation of the theory of formations of groups. Taking formations of groups as the starting point, formations of inverse semigroups are defined, as well as the wider classes of i-formations (i standing for idempotent-separating) and some classes of the kind named f-formations (f standing for fundamental). The relation between the nature of a class of groups and that of certain classes of inverse semigroups with associated groups in the first is discussed. The product of formations is considered, and a product like the Gaschütz's product known for groups is presented for f-formations. [ABSTRACT FROM AUTHOR]

Links between Möbius functions and inverse semigroups are an interesting topic for study. In this paper, we restrict our interest to the submonoid of right units (as a Möbius monoid) of a combinatorial bisimple inverse monoid. The Möbius functions at the end of this paper are Möbius functions of broken Möbius categories (broken up into two parts) via a submonoid and a right monoid action. [ABSTRACT FROM AUTHOR]

Every gauge inverse submonoid (including Jones-Lawson's gauge inverse submonoid of the polycyclic monoid Pn) is a normal submonoid. In 2018, Alyamani and Gilbert introduced an equivalence relation on an inverse semigroup associated to a normal inverse subsemigroup. The corresponding quotient set leads to an ordered groupoid. In this note we shall show that this ordered groupoid is inductive if the normal inverse subsemigroup is a gauge inverse submonoid and the corresponding quotient inverse semigroup by any guage inverse submonoid is isomorphic either to the bicyclic semigroup or to the bicyclic semigroup with adjoined zero. [ABSTRACT FROM AUTHOR]

The symmetric inverse semigroup I(X) on a set X is the collection of all partial bijections between subsets of X with composition as the algebraic operation. We study the minimal Hausdorff inverse semigroup topology on I(X). We present some characterizations of it. When X is countable such topology is Polish. [ABSTRACT FROM AUTHOR]

We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations + and ∘ such that both (S , +) and (S , ∘) are inverse semigroups and a ∘ b + c = a ∘ b - a + a ∘ c and a ∘ a - = - a + a hold, for all a , b , c ∈ S , where - a and a - are the inverses of a with respect to + and ∘ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on S × S . In addition, we provide some methods to construct weak braces. [ABSTRACT FROM AUTHOR]

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an α G -partial Galois extension of S α G and a normal subgroup H of G, we prove that α G induces a unital partial action α G / H of G/H on the subalgebra of invariants S α H of S such that S α H is an α G / H -partial Galois extension of S α G . Second, assuming that G is abelian, we construct a commutative inverse semigroup T par (G , R) , whose elements are equivalence classes of α G -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between T par (G , R) / ρ and T(G, A), where ρ is a congruence on T par (G , R) and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of T par (G , R) reduces to the case where G is cyclic. The set of idempotents of T par (G , R) is also investigated. [ABSTRACT FROM AUTHOR]

We give algebraic characterisations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of its Boolean inverse semigroup of compact open local bisections. It yields in turn algebraic characterisations of both properties for inverse semigroups with meets in terms of subquotients of their Booleanisation. [ABSTRACT FROM AUTHOR]

This is an expository paper in which we explain the basic ideas of topos theory in connection with semigroup theory. We focus mainly on the classifying topos of an inverse semigroup or pseudogroup, and to some extent on creating a dictionary between the language of semigroups and topos theory. We begin with the algebraic theory having to do with inverse semigroups, and then turn to an analysis of pseudogroups using sheaf theory. Our work includes some new material on wide semigroup homomorphisms and their geometric morphisms. [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]

We study properties of eigenvalues of a matrix associated with a randomly chosen partial automorphism of a regular rooted tree. We show that asymptotically, as the level number goes to infinity, the fraction of non-zero eigenvalues converges to zero in probability. [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]