Orthodox semigroups whose idempotents satisfy a certain identity

An orthodox semigroup S is called a left right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy xyx=yx]. Bisimple left right] inverse semigroups have been studied by Venkatesan 6]. In this paper, we clarify the structure of general left right] inverse semigroups. Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx. In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.