By the term semigroup we shall mean a system consisting of a class z of elements, a, b, c, in which there is defined an associative binary operation: a(bc) (ab)c. An ideal of I is a subset S of I such that if a is in 2 and b is in S, then both products ab and ba are in S. Rees(') defines the difference semigroup T = S to be essentially that obtained by collapsing S into a single zero element 0, while the remaining elements of z retain their identity. Thus the T-product of two nonzero elements is defined to be 0 if their I-product lies in S, and otherwise to be the same as defined in 1(2). As in the Schreier theory of group extensions, let us consider the problem of constructing, for given semigroup S and given semigroup T with zero, every possible semigroup z containing S as an ideal, such that z -S is isomorphic with T. Such a z will be called an extension of S by T. If S has a two-sided identity element, every extension of S by T can be obtained by means of a homomorphism of T into S (Theorem 2). If S satisfies the mild A, stated in ?1, every extension of S by T can be obtained by means of a pair of linked ramified homomorphisms of T into the semigroups of left and right translations of S (Theorem 3). Condition A is satisfied if S is what Rees (loc. cit. p. 393) calls a completely simple semigroup without zero. This case is of interest because S is then the Suschkewitsch kernel of I, originally described by Suschkewitsch(3) for finite semigroups, shown by Rees (loc. cit. p. 392) to be the intersection of all the ideals of 1, and further studied by Schwarz(4) and the author(6). In a previous paper(6) the author gave an extension theorem for such an S by a semigroup T having no divisors of zero. Theorem 5 below extends this to arbitrary T.
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