Idempotent semirings with a commutative additive reduct

For every semigroup S , we define a congruence relation ρ on the power semiring (P(S),\cup,\circ) of S . If S is a band, then P(S)/ρ is an idempotent semiring . This enables us to find models for the free objects in the variety of idempotent semiring s whose additive reduct is a semilattice. December 28, 1999     

On quasi completely regular semirings

We extend the concepts of a completely π-regular semigroup and a GV semigroup to semirings and find a semiring analogue of a structure theorem on GV semigroups. We also show that a semiring S is quasi completely regular if and only if S is an idempotent semiring of quasi skew-rings.     

A Combinatorial Property for Semigroups of Matrices

Abstract. In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property . We consider an application of this result to the Burnside problem for groups.     

On semirings whose additive reduct is a semilattice

A semiring S whose additive reduct is a semilattice is called a k-regular semiring if for every a∈S there is x∈S such that a+axa=axa. For a semigroup F, the power semiring P(F) is a k-regular semiring if and only if F is a regular semigroup. An element e∈S is a k-idempotent if e+e ²=e ². Basic properties of k-regular semirings whose k-idempotents are commutative have been studied.     

Hecke Algebras of Group Extensions

Abstract

We describe the Hecke algebra ?(Γ,Γ₀) of a Hecke pair (Γ,Γ₀) in terms of the Hecke pair (N,Γ₀) where N is a normal subgroup of Γ containing Γ₀. To do this, we introduce twisted crossed products of unital *-algebras by semigroups. Then, provided a certain semigroup S ? Γ/N satisfies S ^?1 S = Γ/N, we show that ? (Γ,Γ₀) is the twisted crossed product of ? (N,Γ₀) by S. This generalizes a recent theorem of Laca and Larsen about Hecke algebras of semidirect products.     

Growth of Rank 1 Valuation Semigroups

We consider semigroup S of a rank 1 valuation ? centered on a local ring R. We show that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S. This allows us to give a very simple example of a well ordered subsemigroup of Q+, which is not a value semigroup of a local domain. We also consider the rates of growth which are possible for S. We show that quite exotic behavior can occur. In the final section, we consider the general question of characterizing rank 1 value semigroups.     

Semigroup Rings as Weakly Factorial Domains

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ? be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ?[Γ] is not a weakly factorial domain.     

A Characterization of Semirings Which Are Subdirect Products of a Distributive Lattice and a Ring

E -inversive semiring and a Clifford semiring and show that a semiring S is a subdirect product of a distributive lattice and a ring if and only if S is an E-inversive strong distributive lattice of halfrings. Further a Clifford semiring which is, in fact, an inversive subdirect product of a distributive lattice and a ring, is characterized as a strong distributive lattice of rings. Finally, as a consequence of these results we extend a result of Galbiati and Veronesi [2] in the case of Boolean semirings.     

Prime Ideals of Cancellative Semigroups

Abstract

For a class of prime ideals P of a cancellative semigroup S it is shown that the factors S/P have a structure of a monomial semigroup over a group. Consequences for the semigroup algebras K[S] are discussed.     

On the growth of generating sets for direct powers of semigroups

For a semigroup S its d-sequence is d(S)=(d ₁,d ₂,d ₃,…), where d _i is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.     

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.     

A Structure Preserving Embedding of Semigroups Into Regular Semigroups

For any semigroup S a regular semigroup 𝒞(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schützenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S.     

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.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian rings.     

A Notion of Rank for Right Congruences on Semigroups

ABSTRACT

We introduce a new notion of rank for a semigroup S. The rank is associated with pairs (I,ρ), where ρ is a right congruence and I is a ρ-saturated right ideal. We allow I to be the empty set; in this case the rank of (?, ρ) is the Cantor-Bendixson rank of ρ in the lattice of right congruences of S, with respect to a topology we title the finite type topology. If all pairs have rank, then we say that S is ranked. Our notion of rank is intimately connected with chain conditions: every right Noetherian semigroup is ranked, and every ranked inverse semigroup is weakly right Noetherian.

Our interest in ranked semigroups stems from the study of the class ± b? _S of existentially closed S-sets over a right coherent monoid S. It is known that for such S the set of sentences in the language of S-sets that are true in every existentially closed S-set, that is, the theory T _S of ± b? _S , has the model theoretic property of being stable. Moreover, T _S is superstable if and only if S is weakly right Noetherian. In the present article, we show that T _S satisfies the stronger property of being totally transcendental if and only if S is ranked and weakly right Noetherian.     

Ring semigroups whose subsemigroups intersect

A multiplicative semigroup S is said to be a ring semigroup provided there exists an addition + on S such that (S,+,⋅) is a ring. In this note, we characterize the ring semigroups S with the property that every two nonzero subsemigroups intersect.     

Irreducible Numerical Semigroups Having Toms Decomposition

In this article we prove that if S is an irreducible numerical semigroup and S is generated by an interval or S has multiplicity 3 or 4, then it enjoys Toms decomposition. We also prove that if a numerical semigroup can be expressed as an expansion of a numerical semigroup generated by an interval, then it is irreducible and has Toms decomposition.