On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

Super module amenability of inverse semigroup algebras

In this paper we compare the notions of super amenability and super module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. We find conditions for the two notions to be equivalent. In particular, we study arbitrary module actions of l ¹(E _S ) on l ¹(S) for an inverse semigroup S with the set of idempotents E _S and show that under certain conditions, l ¹(S) is super module amenable if and only if S is finite. We also study the super module amenability of l ¹(S)^?? and module biprojectivity of l ¹(S), for arbitrary actions.     

Module amenability of the second dual and module topological center of semigroup algebras

In this paper we define the module topological center of the second dual $\mathcal{A}^{**}$ of a Banach algebra $\mathcal{A}$ which is a Banach $\mathfrak{A}$ -module with compatible actions on another Banach algebra $\mathfrak{A}$ . We calculate the module topological center of ? ¹(S)^**, as an ? ¹(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ? ¹(S)^** is ? ¹(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.     

(Super) module amenability, module topological centre and semigroup algebras

For a Banach algebra $\mathcal{A}For a Banach algebra A\mathcal{A} which is also an \mathfrakA\mathfrak{A}-bimodule, we study relations between module amenability of closed subalgebras of A"\mathcal{A}', which contains A\mathcal{A}, and module Arens regularity of A\mathcal{A} and the role of the module topological centre in module amenability of A"\mathcal{A}'. Then we apply these results to A=l¹(S)\mathcal{A}=l^{1}(S) and \mathfrakA=l¹(E)\mathfrak{A}=l^{1}(E) for an inverse semigroup S with subsemigroup E of idempotents. We also show that l ¹(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group \fracS ? \frac{S}{\approx}, is amenable. Moreover, we define super module amenability and show that l ¹(S) is super module amenable if and only if \fracS ? \frac{S}{\approx} is finite.     

Module derivation problem for inverse semigroups

The derivation problem for a locally compact group G asserts that each bounded derivation from L ¹(G) to L ¹(G) is implemented by an element of M(G). Recently a simple proof of this result was announced. We show that basically the same argument with some extra manipulations with idempotents solves the module derivation problem for inverse semigroups, asserting that for an inverse semigroup S with set of idempotents E and maximal group homomorphic image G _S , if E acts on S trivially from the left and by multiplication from the right, any bounded module derivation from ? ¹(S) to ? ¹(G _S ) is inner.     

Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ?¹(S) as a Banach right module over ?¹(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c₀(S). For such semigroups S, we also investigate the projectivity of c₀(S). We prove that for many semigroups S for which the Banach algebra ?¹(S) is non-amenable, the ?¹(S)-module ?¹(S) is not injective. The main result about the projectivity of c₀(S) states that for a weakly cancellative inverse semigroup S, c₀(S) is projective if and only if S is finite.     

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ? a congruence on S. Consider the sequence ? ⁽⁰⁾?? ⁽¹⁾???? ⁽ⁿ⁾?? of congruences on S, where ? ⁽⁰⁾=? and, for an arbitrary non-negative integer n, ? ⁽ⁿ⁺¹⁾ is defined by (a,b)∈? ⁽ⁿ⁺¹⁾ if and only if (xa,xb)∈? ⁽ⁿ⁾ for all x∈S. We show that $\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )$ for an arbitrary congruence ? on a semigroup S, where lrc(?) denotes the least left reductive congruence on S containing ?. We focuse our attention on congruences ? on semigroups S for which the congruence $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is left reductive. We prove that, for a congruence ? on a semigroup S, $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is a left reductive congruence of S if and only if $\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}$ is a left reductive congruence on the factor semigroup S/? (here ι _(S/?) denotes the identity relation on S/?). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S ⁿ =S ⁿ⁺¹ is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.     

Certain locally convex topologies on the discrete vector-valued Lebesgue spaces

Let Γ be a set and (E, ‖·‖ _E ) be a nontrivial Banach space. In this paper, through generalizing to vector-valued discrete Lebesgue spaces ? ¹(Γ,E), we show that the topology β ¹(Γ,E) introduced by Singh is, in fact, a type of strict topology. This observation enables us to conclude various basic properties of β ¹(Γ,E). Then, we consider the discrete semigroup algebra ? ¹(S,E) under certain locally convex topologies. As an application of our results, we show that the semigroup algebra (? ¹(S,E), β ¹(S,E)) with the convolution as multiplication is a complete semi-topological (but not topological) algebra.     

On algebraic semigroups and monoids,II

Consider an algebraic semigroup S and its closed subscheme of idempotents, E(S). When S is commutative, we show that E(S) is finite and reduced; if in addition S is irreducible, then E(S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E(S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E(S) is smooth, and its connected components are conjugacy classes of the unit group.     

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.     

Approximate amenability of certain inverse semigroup algebras

In this article, the approximate amenability of semigroup algebra ?¹(S) is investigated, where (S) is a uniformly locally finite inverse semigroup. Indeed, we show that for a uniformly locally finite inverse semigroup (S), the notions of amenability, approximate amenability and bounded approximate amenability of ?¹(S) are equivalent. We use this to give a direct proof of the approximate amenability of ?¹(S) for a Brandt semigroup (S). Moreover, we characterize the approximate amenability of ?¹(S), where S is a uniformly locally finite band semigroup.     

A note on the approximate amenability of semigroup algebras

It is known that the bicyclic semigroup S ₁ is an amenable inverse semigroup. In this note we show that the convolution semigroup algebra ℓ ¹(S ₁) is not approximately amenable.     

The numerical duplication of a numerical semigroup

In this paper we present and study the numerical duplication of a numerical semigroup, a construction that, starting with a numerical semigroup S and a semigroup ideal E?S, produces a new numerical semigroup, denoted by S? ^b E (where b is any odd integer belonging to S), such that S=(S? ^b E)/2. In particular, we characterize the ideals E such that S? ^b E is almost symmetric and we determine its type.     

Second module cohomology group of inverse semigroup algebras

Let S be a commutative inverse semigroup and let E be its subsemigroup of idempotents. In this paper we define the n-th module cohomology group of Banach algebras and we show that H²_l¹(E)(l¹(S),l¹(S)⁽ⁿ⁾)\mathcal {H}^{2}_{\ell^{1}(E)}(\ell^{1}(S),\ell^{1}(S)^{(n)}) is a Banach space for every odd n∈ℕ.     

Weighted Rees matrix semigroup algebras and their applications

In this work, we will describe the weighted semigroup algebra ? ¹(S, ω), where S is a regular Rees matrix semigroup and ω ≥  1. Then as an application, we investigate the amenability of the semigroup algebra ? ¹(S, ω) and its second dual for an arbitrary semigroup S.     

Some results on exclusive sum graphs

Let N(Z) denote the set of all positive integers (integers). The sum graph G ⁺(S) of a finite subset S?N(Z) is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S?N(Z). A sum labelling S is called an exclusive sum labelling if u+v∈S?V(G) for any edge uv∈E(G). We say that G is labeled exclusively. The least number r of isolated vertices such that G∪rK ₁ is an exclusive sum graph is called the exclusive sum number ε(G) of graph G. In this paper, we discuss the exclusive sum number of disjoint union of two graphs and the exclusive sum number of some graph classes.     

Semigroup actions on ordered groupoids

In this paper we prove that if S is a commutative semigroup acting on an ordered groupoid G, then there exists a commutative semigroup S? acting on the ordered groupoid G?:=(G × S)/ρ? in such a way that G is embedded in G?. Moreover, we prove that if a commutative semigroup S acts on an ordered groupoid G, and a commutative semigroup S? acts on an ordered groupoid G? in such a way that G is embedded in S?, then the ordered groupoid G? can be also embedded in G?. We denote by ρ? the equivalence relation on G × S which is the intersection of the quasi-order ρ (on G × S) and its inverse ρ ^?1.     

A class of unary semigroups admitting a Rees matrix representation

In Billhardt et al. (Semigroup Forum 79:101–118, 2009) the authors introduced the notion of an associate inverse subsemigroup of a regular semigroup, extending the concept of an associate subgroup of a regular semigroup, first presented in Blyth et al. (Glasgow Math. J. 36:163–171, 1994). The main result of the present paper, Theorem 2.15, establishes that a regular semigroup S with an associate inverse subsemigroup S ^? satisfies three simple identities if and only if it is isomorphic to a generalised Rees matrix semigroup M(T;A,B;P), where T is a Clifford semigroup, A and B are bands, with common associate inverse subsemigroup E(T) satisfying the referred identities, and P is a sandwich matrix satisfying some natural conditions. If T is a group and A, B are left and right zero semigroups, respectively, then the structure described provides a usual Rees matrix semigroup with normalised sandwich matrix, thus generalising the Rees matrix representation for completely simple semigroups.     

Pseudo-contractibility and pseudo-amenability of semigroup algebras

In this paper, we characterize pseudo-contractibility of ℓ ¹(S), where S is a uniformly locally finite inverse semigroup. As a consequence, we show that for a Brandt semigroup S=M⁰(G,I),{S={\mathcal{M}}^{0}(G,I),} the semigroup algebra ℓ ¹(S) is pseudo-contractible if and only if G and I are finite. Moreover, we study the notions of pseudo-amenability and pseudo-contractibility of a semigroup algebra ℓ ¹(S) in terms of the amenability of S.