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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. I. PRELIMINARIESA

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In this paper the author touches upon two problems, one of which is to determine what is the structure of Cfll) -inversible semigroups and the other is of the general theory of special middle unitary semigroups. The proof of every theorem or lemma, however, is to be stated in another paper. In this paper, we use symbols + or •£ for the direct sum, i.e» the disjoint sum of sets* And moreover let A, 5 be two of any subsets of a semigroup, and use AB for the set { ]

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