On bounded po-semigroups

Abstract

The bounded po-semigroup S is investigated by studying its increasing elements u ( V2). In particular, in S, 01 (= o' lm), 10 (= lmOm), 010 and 101 are all idempotents and 010 = 01 AE 10, 101 = 10 VE 01, E the set of idempotents of S ordered as a subset of S. In 5, Oal = 01 and laO = 10 holds for each a E S. Consequently, S has a zero element z iff 01 = 10 and in that case z = 01. S cannot be cancellative unless it is trivial. JO = 5105 S is the kernel of S and consists of all (idempotents) a E S satisfying aSa = a. Thus when S is a (zero) simple bounded po-semigroup then aSa = (a,z} and either a2 = a or a2 = z for each a E S. When S = X X, the po-semigroup of isotone maps f on the bounded poset X, then JO consists of all constant maps on X, hence JO X. The following generalization of Tarski's fixed point theorem is obtained: Let S be a complete (lattice and a) po-semigroup and let s E S be given. Then the set Es (Jj) of all elements xo E E (E JO resp.) satisfying sxo = s = xo is a nonempty complete lattice when ordered as a subset of S. 1. Let S denote a partially ordered (po) semigroup [1], [3]. Thus S is a semigroup endowed with a partial order <, such that a < b e S implies ac < bc and ca < cb for each c E S. S is bounded if S contains universal bounds 0, 1 such that 0 < s < 1 for each s E S. If S is a po-semigroup which is a complete lattice with respect to < we say that S is a complete posemigroup. An element z(i) satisfying zs = sz = z (is = si = s resp.) for each s E S is a zero (identity resp.) element of the semigroup S [2]. In general, 0, 1 must not be interchanged with z and i. Thus, e.g., let S = Xx [1], be the posemigroup of isotone maps f: X -* X, X a bounded poset. The semigroup operation in Xx is function composition and the order is the pointwise partial order. In Xx we clearly have 01 = 0, 10 = 1. In this paper we study algebraic properties of a po-semigroup S. Two classes of elements, more general than the classes of positive and negative elements [3, p. 154], usually studied in po-semigroups, are introduced and shown to be of special significance. Thus u E S is increasing if u S u2, and v E S is decreasing if v2 < v. Obviously, e is both increasing and decreasing iff e is idempotent, namely e = e2. Let U, V C S denote the sets of increasing (decreasing resp.) elements of S. U, V and E = U n V are ordered as subsets of S. Increasing and decreasing elements of Xx are treated in [6] and shown to form natural extensions of closure and anticlosure operators. As noted in [6, ?7], it is the combined increasing-decreasing character of the identity i Received by the editors May 4, 1975. AMS (MOS) subject classifications (1970). Primary 06A50, 20M 10. C American Mathematical Society 1976