A new notion of semicharacters
Dennison R. Brown, Michael Friedberg · Transactions of the American Mathematical Society · 1969
Introduction.A semigroup S is (uniquely) divisible if, for each xe S, and each positive integer «, there exists (a unique) ye S such that yn = x.In the unique case we write y=xlln.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, "Abelian 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-v = 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