The product of the idempotents and an $\mathcal{H}$ -class of the finite full transformation semigroup

Abstract

We investigate the set products S=EH, where E is the set of idempotents of a finite full transformation semigroup TX and H is an arbitrary \(\mathcal{H}\)-class of TX. We show that S is a semigroup and is a union of \(\mathcal{H}\)-classes of TX. We determine the nature of this union through use of Hall’s Marriage Lemma. We describe Green’s relations and thereby show that S has regular elements of all possible ranks and that \(\operatorname{Reg}(S)\) forms a right ideal of S.