Union Extensions of Semigroupsc)

0. Introduction. The literature on algebraic semigroups contains publications concerning two distinct types of semigroup extensions. Ideal extensions of semigroups were introduced by Clifford in [1] whereas Rédei [9] introduced Schreier extensions of monoids (a monoid is a semigroup containing an identity element). Let A and 5 be two disjoint semigroups and let 5" contain a zero element o. A semigroup L is an ideal extension of A by S if it contains A as ideal and if the Rees factor semigroup E/A is isomorphic with S. In §§4.4 and 4.5 of [2] the early literature on ideal extensions of semigroups is discussed. Recent publications are [5], [7], [8], [11] and [12]. Schreier extensions of monoids are closely related to group extensions. Since Schreier extensions are somewhat complicated to describe and the notions involved are not used in this paper we only give the following references, [13], [3], [4] and [6]. Apart from ideal and Schreier extensions there are other methods of constructing a semigroup L from two given semigroups A and 5 such that it is appropriate to call L an extension of A by S. In §1 we give a general definition of semigroup extensions comprising both ideal and Schreier extensions. A particular class of semigroup extensions, which we call union extensions, is then introduced and considered in some detail. The class of union extensions of a semigroup A by a semigroup S contains the ideal extensions of A by S but is distinct from Schreier extensions of A by S. For the investigation of the existence and for the construction of union extensions of A by S the notion of composition of a semigroup is introduced and worked out in §2. In §3 we describe the method used for investigating union extensions. §4 gives solutions to the questions concerning existence and construction of union extensions of A by S in case S has a single composition. However, one of the cases considered in §4 involves ideal extensions and there we have nothing to add to the existing literature. In §5 the same questions are solved for two cases where 5 has double composition. For the notational conventions and basic concepts used in this paper we refer to [2].