Centralizers of full injective transformations in the symmetric inverse semigroup
The study of centralizers in group theory is a wide topic with enumerable connections. In semigroup theory, the topic has also been considered, in particular within transformation semigroups, for example by Araújo, Konieczny or Lipscomb. Here, contrary to what happens when dealing with permutations, different cases have required particular approaches, depending on being total or partial, injective or not,\dots. In this paper, the author considers total injective maps on a set $X$ within the symmetric inverse semigroup $I(X)$. For each Green relation $S$, the author describes the total injective maps $f$ for which the relation $S$ on its centralizer, the subsemigroup $C(f)$ of $I(X)$, is a restriction of the respective relation $S$ on $I(X)$. Such description is presented in terms of the cycle-ray decomposition of $f$, which generalizes the cycle decomposition of a permutation.
scientific article; zbMATH DE number 6911208
scientific article