SemigroupForumVol. 16(1978) 147-151 RESEARCH ARTICLE ABIAN'S ORDER RELATION FOR SEMIGROUPS C. S. Johnson, Jr. and F. R. McMorris Communicated by Robert McFadden. I. INTRODUCTION. For a ring R, the relation ! defined by a ! b if and only if az = ab is clearly reflexive. Abian [1] notes that < is a partial order on a commutative ring R if and only if R has no non- zero nilpotent elements and Chacron [6] showed that this result is also true for non-commutative rings. Recently this relation has been referred to as "Abian's relation", although Cornish and Stewart [8] point out that Sussman [ii] was probably the first to study it. Some other references to Abian's relation in rings are [2], [3], [4] and [S]. In this note we consider Abian's relation defined on semigroups. We characterize as left separative those semigroups in which it is a partial order and discuss semilattices of groups which are semilat- tices in Abian's order. 2. ABIAN'S ORDER RELATION. Throughout we will let S denote a semigroup with zero and denote Abian'8 ~slation given by a < b if and only if a2 = ab. Recall that S is said to be 8ep~atiue if and only if (i) x2 = xy and y2 = yx imply x = y, and (ii) x2 = yx and y2 = xy imply x = y. If just (i) is satisfied S is called le~ 8epcmative. Abian [i] and Chacron [6] showed that ~ is a partial order on a ring R if and only if R has no nonzero nilpotent elements in which case it also follows that the multiplicative semigroup of R is a partially ordered semigroup. Now it is easy to verify that R has no nonzero nilpotent elements if and only if the multiplicative semigroup of R is separative if and only if the multiplicative semigroup of R is left separative. For semigroups with zero we have separative 147 0037-1912/78/0016-0147$01.00 (~1978 Spdnge~Verlag New York Inc.