Threads without idempotents

Abstract

If a thread S has no idempotents and if S2 = S, then S is iseomorphic with the real interval (0, 1) under ordinary multiplication [2, Corollary 5.6]. Although the result is not nearly as pleasing as the special case just quoted, we shall give here a description of any thread without idempotents. Recall from [1] that a thread is a connected topological semigroup in which the topology is that induced by a total order. First some examples. Let X be a totally ordered set which is a connected space in the interval topology, let T be a subset of X containing, with t, all elements less than t, and let 4 be any continuous function from X into (0, 1) whose restriction, qo, to T is a strictly order-preserving map of T onto (0, a2) where a=l.u.b. 4(X). (We admit that a might be 1.) For such a 4 to exist it is evidently necessary that X not have a least element, that T not have a greatest element and, provided TPX so that the least upper bound, q, of T exists, that +$(q) =a2. If +(X) is the open interval (0, a), define a multiplication in X by: x o y ='0l(4(x)4(y)). With this definition it is quite easy to see that X is a thread without idempotents and that q is a homomorphism. In the event that +(X) is the half closed interval (0, a] (which implies of course that a<1), put A =q-1(a) and B =4-1(a2), observe that q must be the least element of B, and let 41 be any continuous