Bicyclic semigroup

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

Generalized inverse semigroups and amalgamation

Abstract: Given that free products with amalgamation of inverse semigroups exist without “collapse”, or equivalently that any amalgam of inverse semigroups is strongly embeddable in an inverse semigroup, it is natural to ask likewise if free products with amalgamation of generalized inverse semigroups exist without collapse. We note the result of Imaoka [1976b], that free products exist for the class of generalized inverse semigroups. As yet we are unable to answer this question. The main result of this paper is that any amalgam of generalized inverse semigroups is strongly embeddable in a semigroup. Of course this gives some hope that our question above will have an affirmative answer. Our proof is via three representation extension properties. We show that any generalized inverse semigroup has the representation extension property in any containing generalized inverse semigroup; and that any right generalized inverse semigroup has the free (and hence the strong) representation extension property in any containing right generalized inverse semigroup. Our results have been obtained independently by Teruo Imaoka (private communication).

Properties of relatively free inverse semigroups

Abstract: The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if S is combinatorial (i.e., X is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, S does not contain a copy of the bicyclic semigroup) or E-unitary (i.e., E(S) is the kernel of the minimum group congruence) then the relatively free inverse semigroup

Positive matrix semigroups with binary diagonals

Abstract: We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in $${\mathcal{S}}$$ satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.

Decision problems for word-hyperbolic semigroups

Abstract: This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.

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〉.