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

In: Semigroups and Their Applications

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  • Peter R. Jones

    (Marquette University, Department of Mathematics, Statistics and Computer Science)

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〉.

Suggested Citation

  • Peter R. Jones, 1987. "Basis Properties, Exchange Properties and Embeddings in Idempotent-Free Semigroups," Springer Books, in: Simon M. Goberstein & Peter M. Higgins (ed.), Semigroups and Their Applications, pages 69-82, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-3839-7_10
    DOI: 10.1007/978-94-009-3839-7_10
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