The Bergman property for semigroups

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Abstract

In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer-Levi semigroup does not have the Bergman property.

Original languageEnglish
Pages (from-to)212-232
Number of pages21
JournalJournal of the London Mathematical Society
Volume80
Issue number1
Early online date10 Jun 2009
DOIs
Publication statusPublished - Aug 2009

Keywords

  • Finitary power semigroups
  • Generating countable sets
  • Uncountable cofinalities
  • Infinite
  • Transformations
  • Endomorphisms
  • Monoids
  • Ranks

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