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

Abstract

For an arbitrary set X and an equivalence relation \(\mu\) on X, denote by \(P_\mu (X)\) the semigroup of partial transformations \(\alpha\) on X such that \(x\mu \subseteq x(\ker (\alpha ))\) for every \(x\in {{\,\mathrm{dom}\,}}(\alpha )\), and the image of \(\alpha\) is a partial transversal of \(\mu\). Every transversal K of \(\mu\) defines a subgroup \(G=G_{K}\) of \(P_\mu (X)\). We study subsemigroups \(\langle G,U\rangle\) of \(P_\mu (X)\) generated by \(G\cup U\), where U is any set of elements of \(P_\mu (X)\) of rank less than \(|X/\mu |\). We show that each \(\langle G,U\rangle\) is a regular semigroup, describe Green’s relations and ideals in \(\langle G,U\rangle\), and determine when \(\langle G,U\rangle\) is an inverse semigroup and when it is a completely regular semigroup. For a finite set X, the top \(\mathcal {J}\)-class J of \(P_\mu (X)\) is a right group. We find formulas for the ranks of the semigroups J, \(G\cup I\), \(J\cup I\), and I, where I is any proper ideal of \(P_\mu (X)\).

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Acknowledgements

We are grateful to the referee for a very careful reading of the paper.

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Correspondence to Janusz Konieczny.

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Communicated by Victoria Gould.

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André, J., Konieczny, J. Semigroups of partial transformations with kernel and image restricted by an equivalence. Semigroup Forum 102, 10–27 (2021). https://doi.org/10.1007/s00233-020-10116-3

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Keywords

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