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The conjugacy relation is an important tool in the study of group theory and now has been generalized to semigroups in several methods. In particular, the p-conjugacy, o-conjugacy, c-conjugacy, i-conjugacy, n-conjugacy and r-conjugacy in semigroups are introduced and investigated extensively and some problems related to these conjugacy relations are proposed. The aim of this paper is to continue the study of conjugacy relations in semigroups around some of these problems. We first point out that p-conjugacy in a semigroup that is embeddable in a group, is not necessarily transitive. Then we obtain a simple sufficient condition under which the o-conjugacy is the universal relation and prove that this condition is also necessary in some natural classes of semigroups. Finally, we explore some properties of r-conjugacy in restriction semigroups, which extends some results on i-conjugacy in inverse semigroups. Our results have answered or partially answered some problems raised by Araújo, Kinyon, Konieczny and Malheiro in [Proc.Roy.Soc.Edinburgh Sect.A 147, 1169–1214, 2017] and [J.Algebra 533, 142–173, 2019]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate certain submodules in C*-algebras associated to effective étale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C*-algebras are induced by cocycles of étale groupoids if the fixed point algebras contain C*-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras. [ABSTRACT FROM AUTHOR]

AbstractLet P={Xi:i∈I} be a partition of a set X. Denote by T(X) the full transformation semigroup on X, and T(X,P) the subsemigroup of T(X) consisting of all transformations that preserve P . For every subsemigroup S(I) of T(I), let TS(I)(X,P) be the semigroup of all transformations f∈T(X,P) such that χ(f)∈S(I) , where χ(f)∈T(I) defined by iχ(f)=j whenever Xif⊆Xj . We describe regular and idempotent elements in TS(I)(X,P) , and determine when TS(I)(X,P) is a regular semigroup [inverse semigroup]. We characterize Green’s relations on TS(I)(X,P) , describe unit-regular elements in TS(I)(X,P) , and determine when TS(I)(X,P) is unit-regular. [ABSTRACT FROM AUTHOR]

In this paper we consider the questions of which topological semigroups embed topologically into the full transformation monoid ℕ ℕ or the symmetric inverse monoid I ℕ with their respective canonical Polish semigroup topologies. We characterise those topological semigroups that embed topologically into ℕ ℕ and belong to any of the following classes: commutative semigroups, compact semigroups, groups, and certain Clifford semigroups. We prove analogous characterisations for topological inverse semigroups and I ℕ . We construct several examples of countable Polish topological semigroups that do not embed into ℕ ℕ , which answer, in the negative, a recent open problem of Elliott et al. Additionally, we obtain two sufficient conditions for a topological Clifford semigroup to be metrizable, and prove that inversion is automatically continuous in every Clifford subsemigroup of ℕ ℕ . The former complements recent works of Banakh et al. [ABSTRACT FROM AUTHOR]

This paper studies groups and their coset monoids. The anti-abnormal subgroups of a group are firstly introduced and investigated. It is shown that a group is an N ˜ -group if and only if each subgroup of it is anti-abnormal. Also, it is proved that the coset semigroups of N ˜ -groups are exactly the E-reflexive inverse semigroups which are factorisable and the natural connection between their semilattice of idempotents and lattice of subgroups of their group of units is a dual isomorphism. Finally, some characterizations of the coset semigroups of S-groups (U-groups and residually central groups respectively) are given. This extends McAlister's result in 1980. [ABSTRACT FROM AUTHOR]

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 ∪ G ¯) of G, with respect to the extended generating set X ∪ G ¯ , where 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 ∪ G ¯) and leads to a new and simpler proof of the universal property of F(G). [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]

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]

We prove that any set-theoretic solution of the Yang–Baxter equation associated with a dual weak brace is a strong semilattice of non-degenerate bijective solutions. This fact makes use of the description of any dual weak brace S we provide in terms of strong semilattice Y of skew braces B α , with α ∈ Y . Additionally, we describe the ideals of S and study its nilpotency by correlating it to that of each skew brace B α . [ABSTRACT FROM AUTHOR]

Let Γ be a directed graph and Inv (Γ) be the graph inverse semigroup of Γ. Luo and Wang (Semimodularity in congruence lattice of graph inverse semigroups, 2021) 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 (Properties of congruence lattices of graph inverse semigroups, 2022) characterized the directed graph Γ for which is lower semimodular. In the present paper, we show that lower semimodularity, modularity and distributivity in the congruence lattice C (Inv (Γ)) of any graph inverse semigroup Inv (Γ) are equivalent. [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]

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]

Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych C ∗ -algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the C ∗ -algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Basilica group, Gupta–Sidki groups, GGS-groups, multi-edge spinal groups, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk–Erschler group and Fabrykowski–Gupta group) and self-replicating spinal automaton groups. [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.

The conjugacy relation plays an important role in group theory and the conjugacy relation of groups has been generalized to semigroups in various methods by several authors. If aa and bb are elements of a semigroup SS, then aa is called primarily conjugate to bb if a=uva=uv and b=vub=vu for some u,v∈S1u,v\in {S}^{1}. In general, primary conjugacy is reflexive and symmetric, but not transitive. Finding the classes of semigroups in which the primary conjugacy is transitive is an open problem raised by Araújo et al. in the literature. In this article, among other things we prove that the primary conjugacy is transitive in regular ω\omega -semigroups.

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 establish a connection between categorical closedness and nontopologizability of semigroups. In particular, for the class 햳 ퟣ ⁢ 햲 of T 1 topological semigroups we prove that a countable semigroup X with finite-to-one shifts is injectively 햳 ퟣ ⁢ 햲 -closed if and only if X is 햳 ퟣ ⁢ 햲 -discrete in the sense that every T 1 semigroup topology on X is discrete. Moreover, a countable cancellative semigroup X is absolutely 햳 ퟣ ⁢ 햲 -closed if and only if every homomorphic image of X is 햳 ퟣ ⁢ 햲 -discrete. Also, we introduce and investigate a new notion of a polybounded semigroup. It is proved that a countable semigroup X with finite-to-one shifts is polybounded if and only if X is 햳 ퟣ ⁢ 햲 -closed if and only if X is 햳 헓 ⁢ 햲 -closed, where 햳 헓 ⁢ 햲 is the class of Tychonoff zero-dimensional topological semigroups. We show that polybounded cancellative semigroups are groups, and polybounded T 1 paratopological groups are topological groups. [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).

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a 'supertropical trigonometry' and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space R a y (V). In particular, these functions induce a partition of R a y (V) into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars. [ABSTRACT FROM AUTHOR]

A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this, in this paper, we study the class of prefix monoids of finitely presented groups. We obtain a complete description of this class of monoids. All monoids in this family are finitely generated, recursively presented and group-embeddable. Our results show that not every finitely generated recursively presented group-embeddable monoid is a prefix monoid, but for every such monoid, if we take a free product with a suitably chosen free monoid of finite rank, then we do obtain a prefix monoid. Conversely, we prove that every prefix monoid arises in this way. Also, we show that the groups that arise as groups of units of prefix monoids are precisely the finitely generated recursively presented groups, whereas the groups that arise as Schützenberger groups of prefix monoids are exactly the recursively enumerable subgroups of finitely presented groups. We obtain an analogous result classifying the Schützenberger groups of monoids of right units of special inverse monoids. We also give some examples of right cancellative monoids arising as monoids of right units of finitely presented special inverse monoids, and we show that not all right cancellative recursively presented monoids belong to this class.

We consider HNN extensions S * = [ S ; U 1 , U 2 ; ϕ ] where U1 and U2 are inverse monoids of an inverse semigroup S such that, for any u ∈ U i and e ∈ E (S) with u ≥ e in S, there exists f ∈ E (U i) with u ≥ f ≥ e in S, for i ∈ { 1 , 2 } ; we say that U1 and U2 are lower bounded in S. We construct and describe the Schützenberger automata of S * and give conditions for S * to have decidable word problem. Homomorphisms of the Schützenberger graphs of S * are studied and conditions are given for S * to be completely semisimple. When S has decidable word problem and U1 and U2 are finite, we show that S * has decidable word problem. The class of HNN extensions considered here is surprisingly useful and generalizes the class introduced by Jajcayová. A future paper intends to show that any HNN extension of an inverse semigroup can be embedded into an HNN extension where the subsemigroups are lower bounded. [ABSTRACT FROM AUTHOR]

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 for considering quotients of inverse semigroups. 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 quotient of the original universal groupoid. Finally we compute invariant sets and normal subgroupoids induced by special congruences including abelianization. [ABSTRACT FROM AUTHOR]

For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et al., we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra. [ABSTRACT FROM AUTHOR]

For V a vector space over a field, or more generally, over a division ring, it is well-known that every x ∈ End(V ) has an inner inverse; that is, that there exists y ∈ End(V ) satisfying xyx = x. We show here that a large class of such x have inner inverses y that satisfy with x an infinite family of additional monoid relations, making the monoid generated by x and y what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results. P. Nielsen and J. Šter [16] show that a much larger class of elements x of rings R, including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations. A tangential result gives a condition on an endomap x of a set S that is necessary and suffcient for x to have a strong inner inverse in the monoid of all endomaps of S.

In this paper, we study partial group actions on Lie algebras. We describe the structure of the inverse semigroup of all partial automorphisms (isomorphisms between ideals) of a finite-dimensional reductive Lie algebra. Also, we show that every partial group action on a finite-dimensional semisimple Lie algebra admits a globalization, unique up to isomorphism. As a consequence, we obtain that the globalization problem for partial group actions on reductive Lie algebra is equivalent to the globalization problem on its center. [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 present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of 풮 ⁡ (G) {\operatorname{\mathcal{S}}(G)} on the semigroup I ⁢ (X) {I(X)} of partial bijections of X, with 풮 ⁡ (G) {\operatorname{\mathcal{S}}(G)} being an inverse monoid introduced by Exel. We show that any unital premorphism θ : G → S {\theta:G\to S} , where S is an inverse monoid, can be extended to a semigroup homomorphism θ * : T → S {\theta^{*}:T\to S} for any inverse semigroup T with 풮 ⁡ (G) ⊆ T ⊆ P * ⁢ (G) × G {\operatorname{\mathcal{S}}(G)\subseteq T\subseteq P^{*}(G)\times G} , with P * ⁢ (G) {P^{*}(G)} being the semigroup of non-empty subsets of G, and such that E ⁢ (S) {E(S)} satisfies some lattice-theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on Γ ⁢ (X) {\Gamma(X)} , the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X. [ABSTRACT FROM AUTHOR]

We unify various étale groupoid reconstruction theorems such as the following: • Kumjian and Renault's reconstruction from a groupoid C*-algebra; • Exel's reconstruction from an ample inverse semigroup; • Steinberg's reconstruction from a groupoid ring; • Choi, Gardella and Thiel's reconstruction from a groupoid L p {L^{p}} -algebra. We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation ≺ {\prec} on S. The groupoid of ≺ {\prec} -ultrafilters is then isomorphic to the original groupoid G. [ABSTRACT FROM AUTHOR]