Connected ordered topological semigroups with idempotent endpoints. I.

Abstract

This paper is a continuation of an earlier paper(2) of the same title. The semigroups described in the title are called threads. Part I determined the structure of all possible threads with a zero element. The present paper will determine all threads without zero. The numbering of the sections, lemmas, and theorems will continue that of Part I. References to the literature in square brackets will be to the bibliography of Part I. Let S be a thread. Since S is connected and has endpoints, it is compact. Bv a theorem of Numakura(3) and Wallace [11, Lemma 4], S containis a kernel K, i.e. an ideal contained in every ideal of S. K is closed and connected (Wallace [11, Lemma 4]), and so must be a closed subinterval [a, w] of S. K degenerates to a single point (a =cw) if and only if S has a zero; since this case was handled in Part I, we assume throughout Part II that a <w. The algebraic structure of K is one of the two trivial types: either (1) KX=X for all K, XCK, or (2) KX=K for all K, XCK. This is an immediate consequence of Faucett's Theorem 1.3 in [6]; see Lemma 11 below. By passing to the product-dual of S if necessary, we can and shall assume (1). Then K is right-simple, i.e. contains no proper right ideal. Conversely, any right-simple thread has the multiplication (1); indeed, any simple thread coincides with its kernel, anid so has this structure or its product-dual. If J is an ideal of a thread S, and is also a closed interval in S, then we can order the Rees [1(] factor semigroup 7'=S/J in the obvious way, and T becomes thereby a thread with zero. We call S a linear extension of J byT. In particular, any thread S with nondegenerate kernel K is a linear extension of K by 7'= S/IK. Part II is thus concerned with the problem of finding all possible linear extensions S of a right-simple thread K by a thread T with zero. There is a natural division in the results according to whether T is commutative or not. Assume T is commutative. If the zero element of T is at one end, then an extension S of K by T always exists, and all such are given by Theorem 8 below. Let 7'= [rf, e] with f<O<e. If ef(= fe) =0, theii an