On the boundary of algebraic radicals in topological semigroups

Abstract

A topological semigroup is a semigroup with a Hausdorff topology in which multiplication is continuous in both variables. In what follows S will denote a topological semigroup. We let A be any subset of S. The algebraic radical of A is defined to be the set {xCS]xkEA for some integer k_->l} and is denoted by R(A). In [3], K. P. SHUM and C. S. Hoo studied some properties of the algebraic radicals of ideals in compact abelian semigroups. In this paper, we are going to give further investigations on the algebraic radical of an ideal A of S. We find that the boundary of the algebraic radical of an open ideal A of S is closely related with a compact group in S. But in general, the structure of such a compact group is still not known to the author. Here only the existence of a compact group on the boundary of R (A) will be discussed. We list here several definitions which will be frequently used in this paper. Let A be any subset of S, then J(A) = A U SA UASU SAS is called theprincipal ideal generated by A. If S is abelian, then an ideal P of S is said to be prime if ab CP implies that a EP or b CP. An ideal Q of S is said to be primary if ab EQ implies that a E Q or there exists an integer k => 1 such that b k E Q. If S is a semigroup without zero, then S is simple if S does not properly contain an ideal. In an abelian semigroup S, an ideal Q of S is primary if and only if R(Q) is a prime ideal of S, cf. [3]. Throughout, we use cl (C) to denote the closure of the set C, C' for the complement of C and B(C) for the boundary of C. The reader is referred to [4] for other unstated definitions and notations. THEOREM 1. Let S be a connected abelian semigroup with unity and let A be an open ideal of S. Then A is a primqry ideal if and only if the boundary B(R(A)) of R(A) is a subsemigroup of S.