**Authors:** R. S. Gigon

### Abstract

An equivalence relation on a semigroup S is called idempotent pure if all elements of each its class containing idempotent are idempotents, and S is said to be E-disjunctive if the identity relation is the largest idempotent pure congruence on S. We prove that an arbitrary equivalence class of the largest idempotent pure congruence on a semigroup S either consists entirely of E-inversive elements or has no E-inversive elements of S, and then that each E-disjunctive semigroup is necessarily E-inversive. Moreover, in some special classes of semigroups, which are contained in the class of E-inversive semigroups, we investigate the connections between the largest idempotent pure congruence, the least group congruence and some other relation. Basing on this, we give certain new characterizations of the least group (Clifford) congruence.

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