一般化された逆[*]半群の巾等元の恒等式

It is well-known that an orthodox semigroup S is a generalized inverse semigroup if the set E(S) of idempotents of S satisfies one of the following identities : (I.1)x1x2x3x4=x1x3x2x4, (I.2)x1x2x3x1=x1x3x2x1 and (I.3)x1x2x1x3x1=x1x3x1x2x1 (see [4]). In this paper, we shall show that a regular [*-] semigroup S is a generalized inverse [*-] semigroup if E(S) [the set P(S) of projections of S] satisfies the identity (I.2)[(I.1)], but it is not necessarily a generalized inverse [*-] semigroup even if E(S) [P(S)] satisfies (I.3)[(I.2)].