Centralizers of full injective transformations in the symmetric inverse semigroup

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Abstract

Let \(\mathcal {I}(X)\) be the symmetric inverse semigroup of partial injective transformations on a set X (finite or infinite). For \(\alpha \in \mathcal {I}(X)\), let \(C(\alpha )=\{\beta \in \mathcal {I}(X):\alpha \beta =\beta \alpha \}\) be the centralizer of \(\alpha \) in \(\mathcal {I}(X)\). Consider \(\alpha \in \mathcal {I}(X)\) with \({{\mathrm{dom}}}(\alpha )=X\). For each Green relation \(\mathcal {G}\), we determine \(\alpha \) such that \(\mathcal {G}\) in \(C(\alpha )\) is the restriction of the corresponding relation in \(\mathcal {I}(X)\); \(\alpha \) such that all Green relations in \(C(\alpha )\) are the restrictions of the corresponding relations in \(\mathcal {I}(X)\); \(\alpha \) for which \(\mathcal {D}=\mathcal {J}\) in \(C(\alpha )\); \(\alpha \) for which the partial order of \(\mathcal {J}\)-classes in \(C(\alpha )\) is the restriction of the corresponding partial order in \(\mathcal {I}(X)\); and finally \(\alpha \) for which the \(\mathcal {J}\)-classes in \(C(\alpha )\) are totally ordered. The descriptions are in terms of the cycle-ray decomposition of \(\alpha \), which is a generalization of the cycle decomposition of a permutation.

Keywords

Symmetric inverse semigroup Centralizers Green’s relations 

Notes

Acknowledgements

The author is grateful to the referee for a very careful reading of the manuscript and excellent suggestions, which streamlined and improved the paper. This research was supported by a 2014–15 University of Mary Washington Research Grant.

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA

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