Let G, H be groups. G is called an extension of H if there is an epimorphism u: G + H . The congruence Ker u is uniquely determined by the normal subgroups N = o-l( la). Thus we may say that G is an extension of H by N .
Analogously a monoid S is called an extension of a monoid C if there is an epimorphism u: 8 4 > C. By the submonoid B = ~~( 1 ) the congruence Ker u is not uniquely determined. I n 1952 L. R ~D E I [9] described extensions of a monoid in a special case, where the congruence Ker u on S is defined in analogy with a congruence on a group. R. STRECKER ([ll], 1969) and H. GRASSMANN ([2], 1976) generalized the notion of normal submonoids, and investigated the corresponding extension problem. Besides, we can find other notions of normal submonoids in [6] (E. S. LJAPIN) and in [3] (H.
GRASSMANN).
In this paper the notion of a subnormal submonoid is introduced. The SCHREIER extension problem of monoids is considered in the most general case in fi 1, fi 4. I n Β§ 2 the theorem of the prime decomposition for finite monoids is proved. I n fi 3 wreath products of monoids are constructed and GRASSMANN'S theorem [2] about the embedding of associative Schreier products of monoids into wreath products is generalized.
I n Q 5 functorial properties of the functor Ext of central extensions is regarded.
Assume that the semigroups considered in this paper are finite monoids. Some lemmas and theorems will be given without proofs, we refer to [8].
The author is grateful to H. GRASSMANN for his encouragement and helpful advice.
Q 1. Subnormal submonoids and the extension theory of monoids 1.1. Subnormal submonoids Definition 1. Let S be a monoid, B a submonoid of S. B is subnormal if there is a congruence V on S, such that B & V1, where V1 is the %-class which contains the identity 1,. @ is called a B-congruence. A subnormal submonoid B + (1) of S is called nontrivial if there is a B-congruence on S, such that the number of the @-classes is greater than 1. B is trivial subnormal if B = (l} or the only B-congruence is S x S.
Every submonoid B of S is subnormal, since we may choose V = S x S as a B-congruence.
Lemma 1. Let B be a subnormal submonoid of S, 'if a congruence on S. %? i s a B-congruence on S if and only if there are subsets Ai (i = 1, . . ., n) of S with Al 3 1, such that % = (BA,,BA,, ..., BA,,}.