CENTRALISERS IN THE INFINITE SYMMETRIC INVERSE SEMIGROUP
Janusz Konieczny · Bulletin of the Australian Mathematical Society · 2012
Abstract For an arbitrary set $X$ (finite or infinite), denote by $\mathcal {I}(X)$ the symmetric inverse semigroup of partial injective transformations on $X$ . For $ \alpha \in \mathcal {I}(X)$ , let $C(\alpha )=\{ \beta \in \mathcal {I}(X): \alpha \beta = \beta \alpha \}$ be the centraliser of $ \alpha $ in $\mathcal {I}(X)$ . For an arbitrary $ \alpha \in \mathcal {I}(X)$ , we characterise the transformations $ \beta \in \mathcal {I}(X)$ that belong to $C( \alpha )$ , describe the regular elements of $C(\alpha )$ , and establish when $C( \alpha )$ is an inverse semigroup and when it is a completely regular semigroup. In the case where $\operatorname {dom}( \alpha )=X$ , we determine the structure of $C(\alpha )$ in terms of Green’s relations.