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We expand the concept of local embeddability into finite structures (LEF) for the class of semigroups with investigations of non-LEF structures, a closely related generalising property of local wrapping of finite structures (LWF) and inverse semigroups. The established results include a description of a family of non-LEF semigroups unifying the bicyclic monoid and Baumslag–Solitar groups and demonstrating that inverse LWF semigroups with finite number of idempotents are LEF. [ABSTRACT FROM AUTHOR]

Let T(X) (resp. L(V)) be the semigroup of all transformations (resp. linear transformations) of a set X (resp. vector space V). For a subset Y of X and a subsemigroup S (Y) of T(Y), consider the subsemigroup T S (Y) (X) = { f ∈ T (X) : f ↾ Y ∈ S (Y) } of T(X), where f ↾ Y ∈ T (Y) agrees with f on Y. We give a new characterization for T S (Y) (X) to be a regular semigroup [inverse semigroup]. For a subspace W of V and a subsemigroup S (W) of L(W), we define an analogous subsemigroup L S (W) (V) = { f ∈ L (V) : f ↾ W ∈ S (W) } of L(V). We describe regular elements in L S (W) (V) and determine when L S (W) (V) is a regular semigroup [inverse semigroup, completely regular semigroup]. If S (Y) (resp. S (W) ) contains the identity of T(Y) (resp. L(W)), we describe unit-regular elements in T S (Y) (X) (resp. L S (W) (V) ) and determine when T S (Y) (X) (resp. L S (W) (V) ) is a unit-regular semigroup. [ABSTRACT FROM AUTHOR]

Given a set-theoretical solution of the pentagon equation s : S × S → S × S on a set S and writing s (a , b) = (a · b , θ a (b)) , with · a binary operation on S and θ a a map from S into itself, for every a ∈ S , one naturally obtains that S , · is a semigroup. In this paper, we focus on solutions defined in Clifford semigroups S , · satisfying special properties on the set of all idempotents E (S) . Into the specific, we provide a complete description of idempotent-invariant solutions, namely, those solutions for which θ a remains invariant in E (S) , for every a ∈ S . Moreover, we construct a family of idempotent-fixed solutions, i.e., those solutions for which θ a fixes every element in E (S) for every a ∈ S , from solutions given on each maximal subgroup of S. [ABSTRACT FROM AUTHOR]

Constructing concrete examples of certain semigroups could help in implementing algorithms optimized for the users. We give concrete examples of certain finitely presented semigroups, namely Sp,n. Both computational and theoretical approaches are used for studying their structural properties to show that they are quasi-commutative and inverse semigroups. [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]

For a one-to-one partial mapping on an infinite set, we present a criterion in terms of its cycle-chain decomposition that the inverse subsemigroup generated by this mapping is monogenic free inverse. We also give a sufficient condition for a regular rooted tree partial automorphism to extend to a partial automorphism of another regular rooted tree so that the inverse semigroup generated by this extended partial automorphism is monogenic free inverse. The extension procedure we develop is then applied to n-ary adding machines. [ABSTRACT FROM AUTHOR]

A recent paper studied an inverse submonoid M n of the rook monoid, by representing the nonzero elements of M n via certain triplets belonging to Z 3 . In this note, we allow the triplets to belong to R 3 . We thus study a new inverse monoid M ¯ n , which is a supermonoid of M n . We point out similarities and find essential differences. We show that M ¯ n is a noncommutative, periodic, combinatorial, fundamental, completely semisimple, and strongly E ∗ -unitary inverse monoid. [ABSTRACT FROM AUTHOR]

We investigate the computational complexity of various decision problems related to conjugacy in finite inverse semigroups. We describe deterministic algorithms requiring logarithmic space for checking if two elements are conjugate in the full inverse semigroup with respect to various notions of conjugacy. We prove the following two problems are PSPACE -complete: given generators for an inverse semigroup, (1) whether the generated semigroup contains a given idempotent and (2) whether two given elements are ∼ i conjugate in the generated semigroup. We show that checking if an inverse monoid is factorizable is in NC and is NL -hard. We prove that the following problems are all NL -complete: given generators for a partial bijection semigroup, whether the group (1) is nilpotent, (2) is R -trivial, and (3) has central idempotents. We prove that the problem of checking zero membership in a partial bijection semigroup given by generators is L -complete. We also extend several complexity results for partial bijection semigroups to inverse semigroups. [ABSTRACT FROM AUTHOR]

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]

We define a subsemigroup S n of the rook monoid R n and investigate its properties. To do this, we represent the nonzero elements of S n (which are n × n matrices) via certain triplets of integers, and develop a closed-form expression representing the product of two elements; these tools facilitate straightforward deductions of a great number of properties. For example, we show that S n consists solely of idempotents and nilpotents, find the numbers of idempotents and nilpotents, compute nilpotency indexes, determine Green's relations and ideals, and come up with a minimal generating set. Furthermore, we give a necessary and sufficient condition for the jth root of a nonzero element to exist in S n , show that existence implies uniqueness, and compute the said root explicitly. We also point to several combinatorial aspects; describe a number of subsemigroups of S n (some of which are familiar from previous studies); and, using rook n-diagrams, graphically interpret many of our results. [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]

In this study we continue an investigation of the notion of module approximate amenability of a Banach algebra A which is a module over another Banach algebra A. In fact we introduce the class of module boundedly approximately amenable Banach algebras (m.b.app.am.) . It is shown that the class of module boundedly approximately amenable Banach algebra is different from the class of amenable Banach algebras. Also, we show that for an inverse semigroup S with the set of idempotent E, l¹ (S) is module boundedly approximately amenable as l¹ (E)-module if and only if S is amenable. Further examples are given of l¹ -semigroup Banach algebras which are module boundedly approximately amenable but are not amenable. [ABSTRACT FROM AUTHOR]

We provide a close analysis of the connections between pseudogroups, groupoids, and toposes. This analysis provides a topos perspective on both the localic germ groupoid of a pseudogroup defined by Resende, and the topological groupoid of a pseudogroup defined by Lawson and Lenz. In particular, we show how to analyze the topos of a pseudogroup using sheaf theory, leading to an examination of pseudogroup torsors. Consequently we obtain a concrete description of the category of points of an étendue. [ABSTRACT FROM AUTHOR]

We take advantage of the correspondence between pseudogroups and inverse quantal frames, and of the recent description of Morita equivalence for inverse quantal frames in terms of biprincipal bisheaves, to define Morita equivalence for pseudogroups and to investigate its applications. In particular, two pseudogroups are Morita equivalent if and only if their corresponding localic étale groupoids are. We explore the clear analogies between our definition of Morita equivalence for pseudogroups and the usual notion of strong Morita equivalence for C ⁎ -algebras and these lead to a number of concrete results. [ABSTRACT FROM AUTHOR]

Given any amalgam [ S 1 , S 2 ; U ] of inverse semigroups, we show how to construct an amalgam [ T 1 , T 2 ; Z ] such that S 1 ∗ U S 2 is embedded into T 1 ∗ Z T 2 , where S 1 ⊆ T 1 , S 2 ⊆ T 2 , S 1 ∩ Z = S 2 ∩ Z = U and, for any z ∈ Z and h ∈ E (T i) with z ≥ h in T i , where i ∈ { 1 , 2 } , there exists f ∈ E (Z) with z ≥ f ≥ h in T i ; that is, Z is a lower bounded subsemigroup of T 1 and T 2 . A recent paper by the author describes the Schützenberger automata of T 1 ∗ Z T 2 , for an amalgam [ T 1 , T 2 ; Z ] where Z is lower bounded in T 1 and T 2 , giving conditions for T 1 ∗ Z T 2 to have decidable word problem. Thus we can study S 1 ∗ U S 2 by considering T 1 ∗ Z T 2 . As an example, we generalize results by Cherubini, Jajcayová, Meakin, Piochi and Rodaro on amalgams of finite inverse semigroups. [ABSTRACT FROM AUTHOR]

Two semigroups are lattice isomorphic if the lattices of their subsemi-groups are isomorphic, and a class of semigroups is lattice closed if it contains every semigroup which is lattice isomorphic to some semigroup from that class. An orthodox semigroup is a regular semigroup whose idempotents form a sub-semigroup. We prove that the class of all orthodox semigroups in which every nonidempotent element has infinite order is lattice closed. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace , that is a triple (S , + , ⋅) with (S , +) a semigroup and (S , ⋅) an inverse semigroup satisfying the relation a (b + c) = a b + a (a − 1 + c) , for all a , b , c ∈ S , where a − 1 is the inverse of a in (S , ⋅). In particular, we give several constructions of inverse semi-braces which allow for obtaining new solutions of the Yang-Baxter equation. [ABSTRACT FROM AUTHOR]

Given an algebraic action of a semigroup, we construct an inverse semigroup, and we characterize Hausdorffness, topological freeness, and minimality of the associated tight groupoid in terms of conditions on the initial algebraic action. We parameterize all closed invariant subspaces of the unit space of our groupoid, and characterize topological freeness of the associated reduction groupoids. We prove that our groupoids are purely infinite whenever they are minimal, which answers a general open question in the affirmative for our special class of groupoids. In the topologically free case, we prove that the concrete C*-algebra associated with the algebraic action is always a (possibly exotic) groupoid C*-algebra in the sense that it sits between the full and essential C*-algebras of our groupoid. This provides a framework for studying such concrete C*-algebras, allowing us to obtain structural results that were only previously available for very special classes of algebraic actions. For instance, we obtain results on simplicity and pure infiniteness for C*-algebras associated with subshifts over semigroups, actions coming from commutative algebra, and non-commutative rings. These results were out of reach using existing techniques. [ABSTRACT FROM AUTHOR]

In this paper we develop a new groupoid-based structure theory for the class of regular ⁎-semigroups. This class occupies something of a 'sweet spot' between the important classes of inverse and regular semigroups, and contains many natural examples. Some of the most significant families include the partition, Brauer and Temperley-Lieb monoids, among other diagram monoids. Our main result is that the category of regular ⁎-semigroups is isomorphic to the category of so-called 'chained projection groupoids'. Such a groupoid is in fact a triple (P , G , ε) , where: • P is a projection algebra (in the sense of Imaoka and Jones), • G is an ordered groupoid with object set P , and • ε : C → G is a special functor, where C is a certain natural 'chain groupoid' constructed from P. Roughly speaking: the groupoid G = G (S) remembers only the 'easy' products in a regular ⁎-semigroup S ; the projection algebra P = P (S) remembers only the 'conjugation action' of the projections of S ; and the functor ε = ε (S) tells us how G and P 'fit together' in order to recover the entire structure of S. In this way, we obtain the first completely general structure theorem for regular ⁎-semigroups. As a consequence of our main result, we give a new proof of the celebrated Ehresmann–Schein–Nambooripad Theorem, which establishes an isomorphism between the categories of inverse semigroups and inductive groupoids. Other applications will be given in future works. We consider several examples along the way, and pose a number of problems that we believe are worthy of further attention. [ABSTRACT FROM AUTHOR]

Every Grothendieck topos has internal to it a canonical group object, called its isotropy group [Funk et al., 2012]. We continue our investigation of this group, focusing again on locally anisotropic toposes [Funk and Hofstra, 2018]. Such a topos is one admitting an 'etale cover by an anisotropic topos. We present a structural analysis of this class of toposes by showing that a topos is locally anisotropic if and only if it is equivalent to the topos of actions of a connected groupoid internal to an anisotropic topos. In particular we may conclude that a locally anisotropic topos, whence an 'etendue, has isotropy rank at most one, meaning that its isotropy quotient has trivial isotropy [Funk et al., 2018]. [ABSTRACT FROM AUTHOR]

As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in I X , a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse ∧ -semigroups is developed. This class of inverse ∧ -semigroups, otherwise known as 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. [ABSTRACT FROM AUTHOR]

We extend Exel's ample tight groupoid construction to general locally compact étale groupoids in the Hausdorff case. Moreover, we show how inverse semigroups are represented in this way as 'pseudobases' of open bisections, thus yielding a duality which encompasses various extensions of the classic Stone duality. [ABSTRACT FROM AUTHOR]

We prove that the forgetful functor from the category of Boolean inverse semigroups to the category of inverse semigroups with zero has a left adjoint. This left adjoint is what we term the 'Booleanization'. We establish the exact theoretical connection between the Booleanization of an inverse semigroup and Paterson's universal groupoid of the inverse semigroup and we explicitly compute the concrete Booleanization of the polycyclic inverse monoid P n and demonstrate its affiliation with the Cuntz–Toeplitz algebra. [ABSTRACT FROM AUTHOR]

After reviewing aspects of the development of inverse semigroup theory, we describe an approach to studying them which views them as 'non-commutative meet semilattices'. This leads to non-commutative versions of Stone duality and connections with the contemporary theory of étale groupoids and infinite groups analogous to the Thompson groups. [ABSTRACT FROM AUTHOR]

In a group G , elements a and b are conjugate if there exists g ∈ G such that g − 1 a g = 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 ∼ i b , if there exists g ∈ S 1 such that g − 1 a g = b and g b g − 1 = a. The purpose of this paper is to study the conjugacy ∼ i in several classes of inverse semigroups: symmetric inverse semigroups, McAllister P -semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid, stable inverse semigroups, and free inverse semigroups. [ABSTRACT FROM AUTHOR]

Abstract In this paper we give some sufficient and some necessary conditions for an étale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroup algebras and Leavitt path algebras. It turns out that primeness of the algebra is connected with the dynamical property of topological transitivity of the groupoid. We obtain analogous results for semiprimeness. [ABSTRACT FROM AUTHOR]

Abstract Inspired by results for graph C ⁎ -algebras, we investigate connections between the ideal structure of an inverse semigroup S and that of its tight C ⁎ -algebra by relating ideals in S to certain open invariant sets in the associated tight groupoid. We also develop analogues of Conditions (L) and (K) for inverse semigroups, which are related to certain congruences on S. We finish with applications to the inverse semigroups of self-similar graph actions and some relevant comments on the authors' earlier uniqueness theorems for inverse semigroups. [ABSTRACT FROM AUTHOR]

Inverse semigroups and completely regular semigroups share some nice properties and in a certain sense, they are the top success stories in semigroup theory. Therefore a reasonable goal is to find other classes of regular semigroups with the same nice shared properties and try to replicate the successful structure theories achieved for those two classes. Perhaps surprisingly, the semigroup literature is mute on examples of other classes of regular semigroups with the same nice properties, while semigroupists, for decades, voiced the metamathematical conviction of the hopelessness of such a goal. Our guess is that many have tried that approach, but always failed, thus unable to publish (the muteness of the literature) and convinced that the path leads to nowhere (the metamathematical conviction). The aim of this paper is to provide some mathematical content for that metamathematical conviction. In his celebrated theorem, K. Arrow wrote down the nice properties a voting system should have and then went on to prove that no system has those properties. We follow a similar path by writing down the nice common properties of inverse semigroups and of completely regular semigroups, and then show that, under some constraints, any variety having those nice properties is contained in one of the other two (if not in both). The proof of this theorem is by exhaustion: all possible varieties (under certain constraints) are written down, and each one is tested regarding the possession of the nice properties. Many pass the test, but then all of them turn out to be varieties of inverse or completely regular semigroups. This proof requires thousands of lemmas and was only possible because we used ProverX, a system aimed at playing in automated reasoning the same role GAP or MAGMA plays in symbolic computation. The paper ends with some problems for experts in semigroups, equational logic and computer science. [ABSTRACT FROM AUTHOR]

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]

For any graph inverse semigroup G(E) we describe subsemigroups D0 = D ∪ {0} and J0 = J ∪ {0} of G(E) where D and J are arbitrary D-class and J -class of G(E), respectively. In particular, we prove that for each D-class D of a graph inverse semigroup over an acyclic graph the semigroup D0 is isomorphic to a semigroup of matrix units. Also we show that for any elements a, b of a graph inverse semigroup G(E), Ja · Jb ∪Jb · Ja ⊂ J0 b if there exists a path w such that s(w) ∈ Ja and r(w) ∈ Jb. [ABSTRACT FROM AUTHOR]

In this paper, we have studied some properties of inverse semigroups, which play an important role in the semigroup theory. Firstly, we give some general notations, definitions and auxiliary facts related to inverse semigroups. Then we study the close relationship of inverse and regular semigroup and the double inverse semigroup as a special class of inverse semigroups. [ABSTRACT FROM AUTHOR]

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson's amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra 𝒯 2 = A M B , where A and B are Banach algebras (with U -module structure) and M is a Banach A , B -module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca61(6) (2011) 949–958], and they showed that the weak module amenability of 2 × 2 triangular Banach algebra 𝒯 2 (as an ℐ : = α α α ∈ U -bimodule) is equivalent with the weak module amenability of the corner algebras A and B (as Banach U -bimodules). The main aim of this paper is to investigate the module (σ , τ) -amenability and weak module (σ , τ) -amenability of the triangular Banach algebra 𝒯 of order three, where σ and τ are U -module morphisms on 𝒯. Also, we give some results for semigroup algebras. [ABSTRACT FROM AUTHOR]

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over ℓ1(E). This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449-1474, 1995). [ABSTRACT FROM AUTHOR]

Abstract In an earlier work, the author observed that Boolean inverse semigroups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are monoids of generalized rook matrices over groups with zero. (3) There is an order-preserving, one-to-one correspondence between proper varieties of biases and certain finite sequences of varieties of groups, descending in a strong sense defined in terms of wreath products by finite symmetric groups. [ABSTRACT FROM AUTHOR]

Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here, we further modify the Motzkin walks using the elements of a symmetric inverse semigroup as basis states on each step of the walk. This change alters the number of paths allowed in the Motzkin walks and by introducing an appropriate term in the Hamiltonian with a tunable parameter we show that we can jump from a state that violates the area law logarithmically to a state that obeys the area law providing an example of quantum phase transition in a one-dimensional system. [ABSTRACT FROM AUTHOR]

An inverse semigroup S is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if a ∈ S then there exists a unique b ∈ S such that a = a b a and b = b a b. We say that a countable inverse semigroup S is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of S extends to an automorphism of S. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results may be seen as extending both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, such as homogeneous abelian groups and homogeneous finite groups. [ABSTRACT FROM AUTHOR]

In 1959, Philip Hall introduced the locally finite group U, today known as Hall's universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate inU. It can explicitly be described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fraïssé limit of the class of all finite groups. Since its introduction Hall's group and several natural generalisations have been studied widely. In this article we use a generalisation of Fraïssé's theory to construct a countable, universal, locally finite semigroup T, that arises as a direct limit of finite full transformation semigroups, and has the highest possible degree of homogeneity. We prove that it is unique up to isomorphism among semigroups satisfying these properties. We prove an analogous result for inverse semigroups, constructing a maximally homogeneous universal locally finite inverse semigroup I which is a direct limit of finite symmetric inverse semigroups (semigroups of partial bijections). The semigroups T and I are the natural counterparts of Hall's universal group for semigroups and inverse semigroups, respectively. While these semigroups are not homogeneous, they still exhibit a great deal of symmetry. We study the structural features of these semigroups and locate several well-known homogeneous structures within them, such as the countable generic semilattice, the countable random bipartite graph, and Hall's group itself. [ABSTRACT FROM AUTHOR]

It is shown that the projection lattice of a vonNeumann algebra, or more generally every orthomodular lattice X, admits a natural embedding into a group G(X) with a lattice ordering so that G(X) determines X up to isomorphism. The embedding X → G(X) appears to be a universal (non-commutative) group-valued measure on X, while states of X turn into real-valued group homomorphisms on G(X). The existence of completions is characterized by a generalized archimedean propertywhich simultaneously applies to X and G(X). By an extension of Foulis' coordinatization theorem, the negative cone of G(X) is shown to be the initial object among generalized Baer*-semigroups. For finite X, the correspondence between X and G(X) provides a new class of Garside groups. [ABSTRACT FROM AUTHOR]

The class of weakly U-abundant semigroups is an important source of non-regular semi-groups, and it is well studied by semigroup theorists in recent years. An important subclass, called Ehresmann monoids, is deeply investigated by Branco, Gomes and Gould in 2014. In this paper, we are concerned with weakly U-abundant semigroups with strong Ehresmann transversals. Our aim is to give a structure theorem for such semigroups following the standard "Rees Theorem" type approach. As a direct application of the main result, we reobtain the structure theorem of abundant semigroups with quasi ideal adequate transversals by Chen in 2000. [ABSTRACT FROM AUTHOR]

We define and investigate the e-core inverse and f-dual core inverse in a ring with involution as extensions of the core and dual core inverse, respectively. Using these definitions, we present and characterize the e-core partial order and the f-dual core partial order. We describe the sets of all elements a Rickart ∗-ring which are below a given element under the e-core and f-dual core partial order. New characterizations of weighted-EP elements are given too. [ABSTRACT FROM AUTHOR]

In this work we characterize shift spaces over infinite countable alphabets that can be endowed with an inverse semigroup operation. We give sufficient conditions under which zero-dimensional inverse semigroups can be recoded as shift spaces whose correspondent inverse semigroup operation is a 1-block operation, that is, it arises from a group operation on the alphabet. Motivated by this, we go on to study block operations on shift spaces and, in the end, we prove our main theorem, which states that Markovian shift spaces, which can be endowed with a 1-block inverse semigroup operation, are conjugate to the product of a full shift with a fractal shift. [ABSTRACT FROM AUTHOR]

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 it is a topological sum of its H-classes. We describe the structure of Hausdorff locally compact semitopological 0-bisimple inverse ω-semigroups with a monothetic maximal subgroups. We show the following dichotomy: a τ1 locally compact semitopological Reilly semigroup (B(ℤ+, ϑ)0, τ over the additive group of integers ℤ+, with adjoined zero and with a non-annihilating homomorphism ϑ is either compact or discrete. At the end we establish some properties of the remainder of the closure of the discrete Reilly semigroup B(ℤ+, ϑ) in a semitopological semigroup. [ABSTRACT FROM AUTHOR]

We find conditions on topological inverse semigroups X, Y guaranteeing the continuity of any homomorphism h : X → Y having continuous restrictions to any subsemilattice and any subgroup of X. [ABSTRACT FROM AUTHOR]

In this paper, we study inverse semigroups defined on the Bratteli-Vershik systems and SFT covers of 1-solenoids. We show that groupoids of germs of these inverse semigroups are equivalent to the unstable equivalence groupoids of 1-solenoids. Then we prove that Exel's tight $$C^*$$ -algebras of inverse semigroups are strongly Morita equivalent to the unstable $$C^*$$ -algebras of 1-solenoids. [ABSTRACT FROM AUTHOR]