On pseudo-amenability of commutative semigroup algebras and their second duals

In this paper, we first characterize pseudo-amenability of semigroup algebras \(\ell ^1(S),\) for a certain class of commutative semigroups S,  the so-called archimedean semigroups. We show that for an archimedean semigroup S,  pseudo-amenability, amenability and approximate amenability of \(\ell ^1(S)\) are equivalent. Then for a commutative semigroup S,  we show that pseudo-amenability of \(\ell ^{1}(S)\) implies that S is a Clifford semigroup. Finally, we give some results on pseudo-amenability and approximate amenability of the second dual of a certain class of commutative semigroup algebras \(\ell ^1(S)\).     

On generalized inverse transversals

Let S be a regular semigroup, S° an inverse subsemigroup of S.S° is called a generalized inverse transversal of S, if V（x）∩S°≠Ф. In this paper, some properties of this kind of semigroups are discussed. In particular, a construction theorem is obtained which contains some recent results in the literature as its special cases.     

Semilattice indecomposable finite semigroups with large subsemilattices

We show that if Y is a subsemilattice of a finite semilattice indecomposable semigroup S then \({|Y|\leq 2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1}\). We also characterize finite semilattice indecomposable semigroups S which contain a subsemilattice Y with \({|S|=4k+1}\) and \({|Y|=2\left\lfloor \frac{|S|-1}{4} \right\rfloor+1=2k+1}\). They are special inverse semigroups. Our investigation is based on our new result proved in this paper which characterizes finite semilattice indecomposable semigroups with a zero by using only the properties of its semigroup algebra.     

Algebras of Ehresmann semigroups and categories

E-Ehresmann semigroups are a commonly studied generalization of inverse semigroups. They are closely related to Ehresmann categories in the same way that inverse semigroups are related to inductive groupoids. We prove that under some finiteness condition, the semigroup algebra of an E-Ehresmann semigroup is isomorphic to the category algebra of the corresponding Ehresmann category. This generalizes a result of Steinberg who proved this isomorphism for inverse semigroups and inductive groupoids and a result of Guo and Chen who proved it for ample semigroups. We also characterize E-Ehresmann semigroups whose corresponding Ehresmann category is an EI-category and give some natural examples.     

Strict topology on locally compact semigroups

Let S be a locally compact Hausdorff semigroup, and \(\mathfrak {A}\) a solid subalgebra of measure algebra M(S). In this paper, among other results, we find necessary and sufficient conditions on S that implies \(\mathfrak {A}\) is a semi-topological or a topological algebra with respect to the strict topology on M(S). Applications to discrete semigroups, Brandt semigroups and Clifford semigroups are given. An example establishes negatively the open question of Maghsoudi (Semigroup Forum 86:133–139, 2012). Also, we give a correct proof of Proposition 2.1 of Maghsoudi (2012).     

Left Equalizer Simple Semigroups

We characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence \({\varrho}\) on a semigroup S, let \({{\mathbb F}[\varrho]}\) denote the ideal of the semigroup algebra \({{\mathbb F}[S]}\) which determines the kernel of the extended homomorphism of \({{\mathbb F}[S]}\) onto \({{\mathbb F}[S/\varrho]}\) induced by the canonical homomorphism of S onto \({S/\varrho}\). We examine the right colons (\({{\mathbb F}[\varrho] :_{r} {\mathbb F}[S]) = {a \epsilon {\mathbb F}[S] : {\mathbb F}[S]a \subseteqq {\mathbb F}[\varrho]}}\) in general, and in that special case when \({\varrho}\) has the property that the factor semigroup \({S/\varrho}\) is left equalizer simple.     

Ordered groupoid quotients and congruences on inverse semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s \({\mathcal {J}}\)–relation. The corresponding equivalence relation \(\simeq _N\) is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.     

Global determinism of normal orthogroups

If S is a semigroup, the global (or the power semigroup) of S is the set \(\mathcal {P}(S)\) of all nonempty subsets of S equipped with the naturally defined multiplication. A class \(\mathcal {K}\) of semigroups is globally determined if any two semigroups of \({\mathcal {K}}\) with isomorphic globals are themselves isomorphic. We study properties of globals of normal orthogroups and show, in particular, that the class of normal orthogroups is globally determined.     

The principal bundles over an inverse semigroup

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra (Theory Appl Categ 24(7):117–147, 2010). For the topos of sets, we show that torsion-free functors on Loganathan’s category L(S) of an inverse semigroup S are equivalent to a special class of non-strict representations of S, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of S. We describe the correspondence between directed and pullback preserving functors on L(S) and transitive and effective representations of S, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an S-torsor, of an inverse semigroup S in the topos of sheaves \({\mathsf {Sh}}(X)\) on a topological space X. We prove that the category of filtered functors from L(S) to the topos \({\mathsf {Sh}}(X)\) is equivalent to the category of universal representations of S in \({\mathsf {Sh}}(X)\). We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on L(S).     

Pseudo-amenability of certain semigroup algebras

In this paper, we investigate the pseudo-amenability of semigroup algebra ? ¹(S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup \(S={\mathcal{M}}^{0}(G,I)\), the pseudo-amenability of ? ¹(S) is equivalent to the amenability of G.     

Orthodox semigroups and permutation matchings

We determine when an orthodox semigroup S has a permutation that sends each member of S to one of its inverses and show that if such a permutation exists, it may be taken to be an involution. In the case of a finite orthodox semigroup the condition is an effective one involving Green’s relations on the combinatorial images of the principal factors of S. We also characterise some classes of semigroups via their permutation matchings.     

Sets satisfying the Central Sets Theorem

Central subsets of a discrete semigroup S have very strong combinatorial properties which are a consequence of the Central Sets Theorem . We investigate here the class of semigroups that have a subset with zero Følner density which satisfies the conclusion of the Central Sets Theorem. We show that this class includes any direct sum of countably many finite abelian groups as well as any subsemigroup of (?,+) which contains ?. We also show that if S and T are in this class and either both are left cancellative or T has a left identity, then S×T is in this class. We also extend a theorem proved in (Beiglböck et al. in Topology Appl., to appear), which states that, if p is an idempotent in β? whose members have positive density, then every member of p satisfies the Central Sets Theorem. We show that this holds for all commutative semigroups. Finally, we provide a simple elementary proof of the fact that any commutative semigroup satisfies the Strong Følner Condition.     

The One-Sided Inverse Along an Element in Semigroups and Rings

The concept of the inverse along an element was introduced by Mary in 2011. Later, Zhu et al. introduced the one-sided inverse along an element. In this paper, we first give a new existence criterion for the one-sided inverse along a product and characterize the existence of Moore–Penrose inverse by means of one-sided invertibility of certain element in a ring. In addition, we show that \(a\in S^{\dagger }\bigcap S^{\#}\) if and only if \((a^{*}a)^{k}\) is invertible along a if and only if \((aa^{*})^{k}\) is invertible along a in a \(*\)-monoid S, where k is an arbitrary given positive integer. Finally, we prove that the inverse of a along \(aa^{*}\) coincides with the core inverse of a under the condition \(a\in S^{\{1,4\}}\) in a \(*\)-monoid S.     

Generalised domain and <Emphasis Type="Italic">E</Emphasis>-inverse semigroups

A generalised D-semigroup is here defined to be a left E-semiabundant semigroup S in which the \(\overline{\mathcal R}_E\)-class of every \(x\in S\) contains a unique element D(x) of E, made into a unary semigroup. Two-sided versions are defined in the obvious way in terms of \(\overline{\mathcal R}_E\) and \(\overline{\mathcal L}_E\). The resulting class of unary (bi-unary) semigroups is shown to be a finitely based variety, properly containing the variety of D-semigroups (defined in an order-theoretic way in Communications in Algebra, 3979–4007, 2014). Important subclasses associated with the regularity and abundance properties are considered. The full transformation semigroup \(T_X\) can be made into a generalised D-semigroup in many natural ways, and an embedding theorem is given. A generalisation of inverse semigroups in which inverses are defined relative to a set of idempotents arises as a special case, and a finite equational axiomatisation of the resulting unary semigroups is given.     

On algebras of <Emphasis Type="Italic">P</Emphasis>-Ehresmann semigroups and their associate partial semigroups

P-Ehresmann semigroups are introduced by Jones as a common generalization of Ehresmann semigroups and regular \(*\)-semigroups. Ehresmann semigroups and their semigroup algebras are investigated by many authors in literature. In particular, Stein shows that under some finiteness condition, the semigroup algebra of an Ehresmann semigroup with a left (or right) restriction condition is isomorphic to the category algebra of the corresponding Ehresmann category. In this paper, we generalize this result to P-Ehresmann semigroups. More precisely, we show that for a left (or right) P-restriction locally Ehresmann P-Ehresmann semigroup \(\mathbf{S}\), if its projection set is principally finite, then we can give an algebra isomorphism between the semigroup algebra of \(\mathbf{S}\) and the partial semigroup algebra of the associate partial semigroup of \(\mathbf{S}\). Some interpretations and necessary examples are also provided to show why the above isomorphism dose not work for more general P-Ehresmann semigroups.     

Free <Emphasis Type="Italic">n</Emphasis>-Tuple Semigroups

The present paper is devoted to the study of n-tuple semigroups. A free n-tuple semigroup of arbitrary rank is constructed and, as a consequence, singly generated free n-tuple semigroups are characterized. Moreover, examples of n-tuple semigroups are presented, the independence of the n-tuple semigroup axioms is proved, and it is shown that the natural semigroups of the constructed free n-tuple semigroup are isomorphic and the automorphism group of this n-tuple semigroup is isomorphic to a symmetric group.     

Finite derivation type for large ideals

In this paper we give a partial answer to the following question: does a large subsemigroup of a semigroup S with the finite combinatorial property finite derivation type (FDT) also have the same property? A positive answer is given for large ideals. As a consequence of this statement we prove that, given a finitely presented Rees matrix semigroup M[S;I,J;P], the semigroup S has FDT if and only if so does M[S;I,J;P].     

The identities of the free product of two trivial semigroups

We exhibit an example of a finitely presented semigroup S with a minimum number of relations such that the identities of S have a finite basis while the monoid obtained by adjoining 1 to S admits no finite basis for its identities. Our example is the free product of two trivial semigroups.     

New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Raki?, N. ?. Din?i? and D. S. Djordjevi? generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R# with a# = b if and only if (ab)* = ab, ba² = a, and ab² = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.     

Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.