H-両立擬強正則半群について

This is a continuation of the previous papers [3], [4] and [5]. The structure of general quasi-orthodoxsemigroups has been studied in [3] , and construction theorems for upwards directed quasi-orthodox semigroups and split quasi-orthodox semigroups have been given in [5]. On the other hand, the structure of H-compatible orthodox semigroups has been clarified in [4]. In this paper, we shall show that a regular extension S of a completely regular semigroup M～ ∑{M_λ : λ∈Λ} by an inverse semigroup Γ(Λ) is H-compatible if and only if M is H-compatible and Γ(Λ) is H-degenerated. By using this result, it will be also shown that a natural regular semigroup S in the sense of Warne [2] is H-compatible if and only if the union of maximal subgroups of S is an H-compatible subsemigroup of S. Further, a necessary and sufficient condition for a regular semigroup to be H-compatible is also given.