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Pseudo-idempotents in semigroups of functions

The aim of this paper is to generalize Theorem 2.10 (i) of [2]. As stated in [2] this theorem deals with the semigroup of all selfmaps on a discrete space and provides a characterization of H-classes which contain an idempotent. We will generalize this theorem to the case of other semigroups of functions on a discrete space, some semigroups of continuous functions on non-discrete topological spaces, and one semigroup of binary relations. The results in this paper form the main part of chapter 3 of [1]. Some results will be quoted from [1] without proof; the required proofs can easily be supplied by the reader.