Certain homomorphisms of a compact semigroup onto a thread

Abstract

Let S be a compact semigroup and f a continuous homomorphism of S onto the (compact) semigroup T. What can be said concerning the relations among S, f, and T? It is to one special aspect of this problem which we shall address ourselves. In particular, our primary considerations will be directed toward the case in which T is a standard thread. A standard thread is a compact semigroup which is topologically an arc, one endpoint being an identity element, the other being a zero element. The structure of standard threads is rather completely determined e.g. see [20]. Among the standard threads there are three which have a rather special rôle. These are as follows: A unit thread is a standard thread with only two idempotents and no nilpotent element. A unit thread is isomorphic to the usual unit interval [14]. A nil thread again has only two idempotents but has a non-zero nilpotent element. A nil thread is isomorphic with the interval [½, 1], the multiplication being the maximum of ½ and the usual product — or, what is the same thing, the Rees quotient of the usual [0, 1] by the ideal [0,½ ]. Finally there is the idempotent thread, the multiplication being x o y = mm (x, y). These three standard threads can often be considered separately and, in this paper, we reserve the symbols I1I2 and I3 to denote the unit, nil and idempotent threads respectively. Also, throughout this paper, by a homomorphism we mean a continuous homomorphism.