Showing papers on "Bicyclic semigroup published in 1987"

The minimal faithful degree of a fundamental inverse semigroup

Abstract: This paper shows that the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.

Analogues of the bicyclic semigroup in simple semigroups without idempotents

Abstract: By an important theorem of Andersen, any semigroup, containing idempotents, which is simple but not completely simple contains a copy of the bicyclic semigroup B = 〈a, b | ab = 1〉. In this paper the semigroups A = 〈a, b | a2b = a〉 and C = 〈a, b | a2b = a, ab2 = b〉 are shown to play a similar role in various classes of simple semigroups without idempotents, particularly in those for which Green's relation is nontrivial. For example it is shown that every right simple semigroup without idempotents is a union of copies of A; every finitely generated simple semigroup without idempotents contains either A or C. In a generalisation of a different sort it is shown that the bicyclic semigroup divides every simple semigroup without idempotents.Similar results are obtained for 0-simple semigroups without nonzero idempotents.

Center of the semigroup algebra of a finite inverse semigroup over the field of complex numbers

Abstract: In the semigroup algebra A of a finite inverse semigroup S over the field of complex numbers to an indempotent e there is assigned the sum σ(e)=e+∑(−1)KeL1⋯eiK, where ei,...,em are maximal preidempotents of the idempotent e, and the summation goes over all nonempty subsets {i1,...,ik} of the set {1,...m} Then for any class K of conjugate group elements of the semigroup S the element K=∑a·(a−1a) (the summation goes over all a∈g) is a central element of the algebra A, and the set {K} of all possible such elements is a basis for the center of the algebra A.

Basis Properties, Exchange Properties and Embeddings in Idempotent-Free Semigroups

Abstract: Two “basis properties” are considered for semigroups and associated algebras. These were introduced by the author to study inverse semigroups and groups, motivated by well-known properties of vector spaces. First these basis properties are studied in the abstract, from the point of view of exchange properties; then those semigroups with the “strong” basis property are determined for many classes of semigroups, including all regular and all periodic semigroups. The main method of proof is to eliminate undesirable types of semigroups by showing that each contains a certain special sub-semigroup, for instance the bicyclic semigroup. This prompts the study of analogs of a theorem of Andersen on embeddings of the bicyclic semigroup: such analogs are found for the semigroups A = 〈a,b|a2b = a〉 and C = 〈a,b|a2b = a,ab2 = b〉.

A Class of Inverse Semigroup Algebras

Abstract: In 1976, Domanov showed that the algebra of an inverse semigroup S over a field F is semiprimitive (that is, has zero Jacobson radical) if the algebra of each maximal subgroup of S over F is semiprimitive. It is known that the converse statement is false in general. The principal purpose of this paper is to announce that if the semi-lattice of S satisfies a certain finiteness condition, introduced by Teply, Turman and Quesada in 1980, then the converse does hold. Corresponding results for primitivity are also discussed.

The Identity Element in Inverse Semigroup Algebras

Abstract: W. D. Munn and R. Penrose (1957) obtained an explicit formula for the identity element 1A of the semigroup algebra A of an arbitrary finite inverse semigroup S. An alternative (inductive) characterization of 1A is presented, giving new information about how the form of 1A depends on the Vagner-Preston order in S.