Derived rees matrix semigroups as semigroups of transformations

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Abstract

An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that TX contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that TX contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.

Original languageEnglish
Pages (from-to)219-233
Number of pages15
JournalSemigroup Forum
Volume73
Issue number2
DOIs
Publication statusPublished - Oct 2006

Keywords

  • Regular Semigroup
  • Semigroup Forum
  • Transformation Semigroup
  • Rees Matrix Semigroup
  • Dual Isomorphism

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