Semigroups of partial transformations with kernel and image restricted by an equivalence

Jorge M. André, Janusz Konieczny

Research output: Contribution to journalArticle

Abstract

For an arbitrary set X and an equivalence relation μ on X, denote by Pμ(X) the semigroup of partial transformations α on X such that xμ⊆x(ker(α)) for every x∈dom(α), and the image of α is a partial transversal of μ. Every transversal K of μ defines a subgroup G=GK of Pμ(X). We study subsemigroups ⟨G,U⟩ of Pμ(X) generated by G∪U, where U is any set of elements of Pμ(X) of rank less than |X/μ|. We show that each ⟨G,U⟩ is a regular semigroup, describe Green’s relations and ideals in ⟨G,U⟩, and determine when ⟨G,U⟩ is an inverse semigroup and when it is a completely regular semigroup. For a finite set X, the top J-class J of Pμ(X) is a right group. We find formulas for the ranks of the semigroups J, G∪I, J∪I, and I, where I is any proper ideal of Pμ(X).
Original languageEnglish
JournalSemigroup Forum
DOIs
Publication statusPublished - 3 Jul 2020

Keywords

  • Partial transformation semigroups
  • Equivalence relations
  • Green’s relations
  • Regular semigroups
  • Ideals
  • Rank

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