Extensions of Brandt semigroups and applications
R. J. Warne · Illinois Journal of Mathematics · 1966
Clifford gave a general means of finding all possible extensions of a (weakly reductive) semigroup S by a semigroup T with zero [2].However, as in group theory, it is generally difficult to give an explicit determination of the exten- sions for special types of semigroups.This has been done for only two cases:(1) S completely simple and T arbitrary.(2) S a group and T a completely 0-simple semigroup [2].The first is due to Clifford [1] and the second to Munn[2].In [3], Warne determined when the extensions of a completely 0-simple semigroup by a completely 0-simple semigroup are determined by a partial homomorphism.The main result of this paper is the determination of all extensions of a Brandt semigroup by an arbitrary semigroup.We first use this theorem to determine when an extension of a Brandt semigroup by a regular 0-bisimple semigroup is given by a partial homomorphism.We then use the theorem to find the number of extensions of a Brandt semigroup by a simple group (with zero) in a certain case.Let S and T be disjoint semigroups, T having a zero element 0. A semi- group V will be called an (ideal) extension of S by T if it contains S as an ideal, and if the Rees factor semigroup V/S [1] is isomorphic with T.Let V be an extension of a semigroup S by a semigroup T with zero; we will use the following notations.If S has a zero, it is denoted by 0 (0 is then automatically the zero of V).The zero of T is denoted by 0'.Multiplica- tion in V is denoted by o, while multiplication in S or T is denoted simply by iuxtaposition.The elements of S are denoted by lower case and the elements of T by capital roman letters.The set of non-zero elements of any semigroup P with zero is denoted by P*.If V is an extension of S by T (with zero) we say that V is determined by a partial homomorphism if there exists a partial homomorphism T* --S such that for all A, B T*, c, d e S AoB =AB (A-)(B-) Aoc (A)c; if AB O if AB 0'; coA c(A); cod cd.