A new notion of semicharacters

Abstract

0. Introduction. A semigroup S is (uniquely) divisible if, for each x E S, and each positive integer n, there exists (a unique) y E S such that yn = x. In the unique case we write y=X1ln. Uniquely divisible commutative semigroups, referred to in the sequel as UDC semigroups, have been characterized in [7]. Compact topological semigroups satisfying this hypothesis have been studied in [6], [11], [12], and [13]. Material of a related nature occurs in [15] and [17]. In [6], the authors showed that if S is a finite-dimensional compact UDC semigroup in which the set of idempotents is totally disconnected, then there exist sufficiently many continuous homomorphisms (semicharacters) of S into the complex unit disk to separate points. This usage of the complex disk as a range space is in line with the classical philosophy, group is to the circle group as Abelian semigroup is to the complex unit disk semigroup. A large amount of work has been done on the investigation of this analogy; in particular, see [9] for a comprehensive survey of the algebraic results in this direction. For the case of Abelian topological semigroups and continuous semicharacters, see [16] and [23]. However, it is clear that the idempotent structure of the complex disk makes it unsuitable as a range space for continuous homomorphisms on general Abelian topological semigroups. In particular, if S is a connected topological semilattice, then any continuous homomorphism of S into the complex disk must be trivial. This deficiency is well known, and reasonable substitutes for the complex disk have been sought for some time. If each maximal group of S is trivial, then one of the most appealing replacements for the disk is some form of thread -a semigroup on a space homeomorphic to the unit interval in which one endpoint acts as an identity and the other as a zero. The complete structure of threads is given in [21], and questions concerning their suitability as range spaces for continuous semicharacters are raised therein. The basic building blocks for threads are U, the interval [0, 1] under real number multiplication; M, the interval [0, 1] under multiplication xy =min {x, y}; and C, the interval [1/2, 1] under multiplication x y = max {1 /2, xy}, where xy represents the ordinary real number product of x and y. The semigroup U has been used successfully as a range space for a certain class of compact UDC semigroups in [6]. The semigroup M is a very logical range space for the category of compact topological semilattices; the problem of whether every