Second centralizers in the semigroup of injective transformations

Abstract

For an element [Formula: see text] of a semigroup [Formula: see text], denote by [Formula: see text] and [Formula: see text], respectively, the first and second centralizers of [Formula: see text] in [Formula: see text]. Let [Formula: see text] be an arbitrary set and let [Formula: see text] be the semigroup of all injective mappings from [Formula: see text] to [Formula: see text]. For an arbitrary [Formula: see text], we characterize the elements of [Formula: see text], show that [Formula: see text] is a direct product of finite cyclic groups and the infinite cyclic monoid, and determine Green’s relations in [Formula: see text]. We also describe those [Formula: see text] for which [Formula: see text], which will provide a class of maximal commutative subsemigroups of [Formula: see text].