GENERALIZA nON OF CONTINUOUS POSETS

In this paper we develop a general theory of continuity in partially ordered sets. Among the interesting special cases of this theory is the theory of continuous lattices developed by D. Scott, J. Lawson and others. Introduction. In this paper we are going to generalize the concept of a continuous poset, using the theory of Galois connections. In this sense we are following Hofmann and Stralka who in [8] first used this technique in connection with continuous lattices. In [19] Wagner, Thatcher and Wright developed a "uniform approach" method which enable them to generalize the concept of an algebraic poset. At the end of that article they suggested that continuous posets might be investigated by the same method. This paper is a response to their suggestion. An essential idea of this article is based on the following two facts. (1) A lattice (or poset) P is a complete lattice if and only if the function x ->.j, x from P to the ideal completion of P has a right adjoint sending the ideal I to sup(l). (2) This right adjoint has itself a right adjoint (sending x to * x) if and only if the lattice (or poset) P is continuous. We use the existence of these three functions (for arbitrary extensions of posets) as our definition of continuity. § 1 introduces the main tool which we use in this thesis, the theory of Galois connections. In §2 we present the main concept, that of a continuous extension, and develop some basic results concerning it. In particular, we show that a continuous extension naturally gives rise to a dually residuated closure operator (which, in tum, determines the continuous extension up to isomorphism). We also show that in some special cases the concepts of continuity and distributivity coincide. Some results in this part of the article overlap with the work done in [1,6]. The third section is devoted to a general study of well below relations which helps us to construct examples of continuous extensions. This part is closely related to the section about auxiliary relations in [7]. The next section is mainly devoted to the construction of two operators associated with a given continuous extension. One of these is a closure operator, the other is an anticlosure operator. One of the main theorems of this paper is Theorem 4.18 which leads the reader to the notion of strong continuity. We show that a continuous poset Received by the editors December 24, 1980 and, in revised form, April 26, 1981. 1980 Mathematics Subject Classification. Primary 06AlO; Secondary 68A05.