Centralizers of full injective transformations in the symmetric inverse semigroup

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.