Regular centralizers of idempotent transformations
André, Jorge M.; Araújo, João; Konieczny, J.
Semigroup Forum, 82(2) (2011), 307-318
http://dx.doi.org/10.1007/s00233-010-9274-6
Denote by T (X) the semigroup of full transformations on a set X. For ? ? T (X), the centralizer of ? is a subsemigroup of T (X) defined by C(?) = {? ? T (X) : ?? = ??}. It is well known that C(idX) = T (X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(idX) contains all non-invertible transformations in C(idX).
This paper generalizes this result to C(?), an arbitrary regular centralizer of an idempotent transformation ? ? T (X), by describing the subsemigroup generated by the idempotents of C(?). As a corollary we obtain that the subsemigroup generated by the idempotents of a regular C(?) contains all non-invertible transformations in C(?) if and only if ? is the identity or a constant transformation.
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