Bicyclic semigroup

About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.

Locally adequate semigroup algebras

Abstract: Abstract We build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant 0-J*$0{\\rm{ - }}{\\cal J}*$-simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the ℛ*${\\cal R}*$-classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.

Cross-connections of completely simple semigroups

Abstract: A completely simple semigroup S is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that S is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup. In the study of structure theory of regular semigroups, K.S.S. Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left(right) ideals using cross-connections. A normal category C is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories C and D is a local isomorphism frm D to N*C where N*C is the normal dual of the category C. In this paper, we identify the normal categories associated with a completely simple semigroup S and show that the semigroup of normal cones TL(S) is isomorphic to a semi-direct product of semigroups. We characterize the cross-connections in this case and show that each sandwich matrix P correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Abstract: Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number

The free envelope of a finitely generated commutative semigroup

Abstract: Let S be a commutative semigroup. A quasi-universal free semigroup of S is a free commutative semigroup with identity F together with a homomorphism -q of S into F such that any homomorphism of S into a free commutative semigroup with identity factors through -q; if there is uniqueness in this factorization, we say that (F, -q) is a universal free semigroup of S. If S is finitely generated, there exists a "smallest" quasi-universal free semigroup of S; we call it the free envelope of S. Its construction and study is the first object of this paper, the second being the application of the free envelope to the study of cancellative and power-cancellative commutative semigroups. We construct the free envelope in the first section. Cancellative and powercancellative semigroups appear in ?2; we prove that a finitely generated commutative semigroup is embeddable into a free commutative semigroup with identity if and only if it has these properties and has either no identity or a trivial group of units; then it is embeddable into its free envelope. The study of this latter embedding gives, conversely, a number of interesting properties of the free envelope in the general situation. The dual of a finitely generated commutative semigroup S may be defined as the semigroup S* of all homomorphisms of S into the additive semigroup of all nonnegative integers, under pointwise addition. Using free envelopes, we prove in ?3 that S***-S* and investigate the relationship between S and S**. This yields in turn a number of results concerning universal free semigroups when they exist. A study of various dimensions completes the section. ?4 deals with embeddings Sc T such that every relation which holds in S can be deduced from the presentation of S in T without using any relation which may hold in T (in which case we say that T kills S). We show that, if S is finitely generated, the (inclusion) homomorphism of S into T can be extended to the free envelope of S; if furthermore T is cancellative, power-cancellative and without identity element, then T contains subsemigroups which kill S and are minimal with that property. This paper has benefited from numerous suggestions, by the members of the Tulane semigroup seminar, especially William R. Nico and A. H. Clifford, and by our referee, which we acknowledge gladly.