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Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup

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Abstract

We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a cycle. Finally, as an application, we use involution matchings to prove that when the base set has at least four members, a finite full transformation semigroup has no cover by inverse subsemigroups that is closed under intersection.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ, UK

    Peter M. Higgins

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  1. Peter M. Higgins
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Correspondence to Peter M. Higgins.

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Communicated by Norman R. Reilly.

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Higgins, P.M. Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup. Semigroup Forum 98, 669–689 (2019). https://doi.org/10.1007/s00233-019-10018-z

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  • DOI: https://doi.org/10.1007/s00233-019-10018-z

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