Threads without Idempotents

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 [l] 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 £ be a subset of X containing, with t, all elements less than /, and let 4> be any continuous function from X into (0, 1) whose restriction, (X) is the open interval (0, a), define a multiplication in X by: x o y = o~l(4>(x) is a homomorphism. In the event that ~1(a) and B=<b~1(a2), observe that q must be the least element of B, and let \p be any continuous