On Kiselman’s semigroup

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On Kiselman’s semigroup

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by Ganna Kudryavtseva , Volodymyr Mazorchuk

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BibTeX

@MISC{Kudryavtseva_onkiselman’s,
    author = {Ganna Kudryavtseva and Volodymyr Mazorchuk},
    title = {On Kiselman’s semigroup},
    year = {}
}

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Abstract

We study the algebraic properties of the series Kn of semigroups, which is inspired by [Ki] and has origins in convexity theory. In particular, we describe Green’s relations on Kn, prove that there exists a faithful representation of Kn by n × n matrices with non-negative integer coefficients (and even explicitly construct such a representation), and prove that Kn does not admit a faithful representation by matrices of smaller size. We also describe the maximal nilpotent subsemigroups in Kn, all isolated and completely isolated subsemigroups, all automorphisms and anti-automorphisms of Kn. Finally, we explicitly construct all irreducible representations of Kn over any field and describe primitive idempotents in the semigroup algebra (which we prove is basic). 1

Keyphrases

kiselman semigroup faithful representation green relation series kn non-negative integer coefficient semigroup algebra algebraic property isolated subsemigroups maximal nilpotent subsemigroups describe primitive idempotents irreducible representation convexity theory

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