Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency.     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency. (Received 17 April 2001; in revised form 15 September 2001)     

Every finite semigroup is embeddable in a finite relatively free semigroup

The title result is proved by a Murskii-type embedding.Results on some related questions are also obtained. For instance, it is shown that every finitely generated semigroup satisfying an identity ξ^d=ξ^2d is embeddable in a relatively free semigroup satisfying such an identity, generally with a larger d; but that an uncountable semigroup may satisfy such an identity without being embeddable in any relatively free semigroup.It follows from known results that every finite group is embeddable in a finite relatively free group. It is deduced from this and the proof of the title result that a finite monoid S is embeddable by a monoid homomorphism in a finite (or arbitrary) relatively free monoid if and only if its group of invertible elements is either {e} or all of S.     

Retracts and monomorphism monoids in semigroup varieties

If a semigroup varietyV contains the variety of commutative three-nilpotent semigroups, or if it is a variety of bands containing all semilattices, then, for anyA∈V and any left cancellative monoidM, there is a semigroupS∈V such thatA is a retract ofS andM is isomorphic to the monoid of all injective endomorphisms ofS.     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.     

Principal Weak Flatness and Regularity of Diagonal Acts

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ? left P(P) ? weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 _X , D(S) is regular.     

Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable

For a semigroup S, the set of all isomorphisms between the subsemigroups of the semigroup S with respect to composition is an inverse monoid denoted by PA(S) and called the monoid of local automorphisms of the semigroup S. The semigroup S is called permutable if, for any couple of congruences ρ and σ on S, we have ρ ∘ σ = σ ∘ ρ. We describe the structures of a finite commutative inverse semigroup and a finite bundle whose monoids of local automorphisms are permutable.     

On Finite Presentability of Direct Products of Semigroups

A finite semigroup S is said to preserve finite generation (resp., presentability) in direct products, provided that, for every infinite semigroup T, the direct product S × T is finitely generated (resp., finitely presented) if and only if T is finitely generated (resp., finitely presented). The main result of this paper is a constructive necessary and sufficient condition for S to preserve both finite generation and presentability in direct products. The condition is that certain graphs, (s), one for each s S, are all connected. The main result is illustrated in three examples, one of which exhibits a 4-element semigroup that preserves finite generation but not finite presentability in direct products.1991 Mathematics Subject Classification: 20M05, 05C25The first author is financially supported by the Sub-Programa Ciência e Tecnologia do 2° Quadro Comunitário de Apoio (grant number BD/ 15623/98). The author also acknowledges the support of the Centro de Álgebra da Universidade de Lisboa and of the Projecto Praxis 2/2.1/MAT/73/94. The second author acknowledges partial financial support from the Nuffield Foundation.     

Quasi-duality for the rings of generalized power series *

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered commutative monoid which is also artinian. For any bimodule _AM_B , we construct a bimodule _A[[S]]M[S]_B[[S]] and prove that _AM_B defines a quasi-duality if and only if the bimodule _A[[S]]M[S]_B[[S]] defines a quasi-duality. As a corollary, it is shown that if a ring A has a quasi-duality then the ring A[[S]] of generalized power series over A has a quasi-duality.     

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

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.     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

On Automatic Rees Matrix Semigroups

We consider a Rees matrix semigroup S = M[U; I, J; P] over a semigroup U, with I and J finite index sets, and relate the automaticity of S with the automaticity of U. We prove that if U is an automatic semigroup and S is finitely generated then S is an automatic semigroup. If S is an automatic semigroup and there is an entry p in the matrix P such that pU ¹ = U then U is automatic. We also prove that if S is a prefix-automatic semigroup, then U is a prefix-automatic semigroup.     

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].     

Finite basis problem for 2-testable monoids

A monoid S ¹ obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S ¹ is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.     

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R., _S P _R,R Q _S ,〈〉 , ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζ _S = {(s ₁, s ₂) ∈S×S|ss ₁ = ss ₂, ∀s∈S}, S' = S/ζ _S and US-FAct = { _S M∈S− Act |SM = M and SHom _S (S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom _S (S, ∐ _i∈I S) →∐ _i∈I S, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism. The research is partially supported by a UGC(HK) grant #2160092. Project is supported by the National Natural Science Foundation of China     

E-special Ordered Regular Semigroups

An ordered regular semigroup S is E-special if for every x ∈ S there is a biggest x ⁺ ∈ S such that both xx ⁺ and x ⁺ x are idempotent. Every regular strong Dubreil–Jacotin semigroup is E-special, as is every ordered completely simple semigroup with biggest inverses. In an E-special ordered regular semigroup S in which the unary operation x → x ⁺ is antitone the subset P of perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal of P if and only if S has a biggest idempotent. If S ⁺ is a subsemigroup and S does not have a biggest idempotent, then P contains a copy of the crown bootlace semigroup.     

Semigroups with magnifiers admitting minimal subsemigroups

An element a of a semigroup S is a left magnifier if λ_a, the inner left translation associated with a, is surjective and is not injective (E. S. Ljapin [11]). When this happens there exists a proper subset M of S such that the restriction to M of λ_a is bijective. In that case M is said to be a minimal subset for the left magnifier a (F. Migliorini [13], [14], [15]). Remark that if S is a semigroup having left identities then every left magnifier of S admits minimal subsets which are right ideals. Characterisations for semigroups with left magnifiers which also contain left identities have been given by E. S. Ljapin and R. Desq, using the bicyclic monoid. The general problem, precisely to give a characterization of semigroups having left magnifiers, is still open.     

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.