In § 1 of this paper, we extend the notions defined in [1] and [2], and note some relatiofiships among them. In §2, we give characterizations of primary semigroups and give a generalization of a theorem of A.Anjaneyulu [1, Theorem 2]. And in §3, we treat completely π-regular elementsin semigroups and extend the results obtained in [2
] and [3].