## GENERALIZATION OF CONTINUOUS POSETS

### Abstract (summary)

The theory of continuous lattices was introduced by D. Scott in the early 1970's as the foundation for a mathematical theory of computation. After the appearance of Scott's papers, other authors suggested ways of extending and generalizing Scott's ideas. The present work is intended as a further contribution to the development of this body of theory.

Let P and Q be partially ordered sets. A pair of order preserving maps (f,g), g:Q (--->) P and f:P (--->) Q, is called a Galois connection from Q to P if for each x (ELEM) Q and each y (ELEM) P, f(g(x)) (LESSTHEQ) x and g(f(y)) (GREATERTHEQ) y. If this condition holds we say that the map f is the right adjoint of the map g, and the map g is the left adjoint of the map f. An order preserving map from a partially ordered set to a partially ordered set is called residuated (dually residuated) if it has a left adjoint (right adjoint).

If P and Q are partially ordered sets and (f,g) is a Galois connection from Q to P, then the map f is residuated. If we further assume that the map f is dually residuated and maps Q onto P, then we define the triple (P,Q,g) to be a continuous extension.

The notion of continuous extension provides a unifying concept for various parts of the theory of partially ordered sets. In particular, the following structures are closely associated with special cases of continuous extensions: (1)Continuous posets (and thus, in particular, continuous lattices). (2)Completely distributive lattices. (3)Lattices in which every element is a finite join of coprime elements.

Some of my research concerns the characterization of two interesting subclasses of the class of continuous extensions: the class of M-continuous extensions and the class of strongly continuous extensions. We have shown that the three special cases mentioned above belong to both subclasses. We have also shown that every strongly continuous extension gives rise to a completely distributive lattice. Using this result we obtain several applications to the theory of continuous lattices. For example:

Theorem. Let P be a chain-complete poset and let L(P) denote the family

of all lower subsets of P. Then the following conditions are equivalent:

(1)P is a continuous poset. (2)For every family {S(,i) (VBAR) i (ELEM) I} (L-HOOK EQ) L(P),(, )

(INTERSECT){S(,i) (VBAR) i (ELEM) I} = (INTERSECT){S(,i) (VBAR) i (ELEM) I}. (Here S(,i) denotes the Scott closure of(, )

S(,i).) (3)For every Scott closed set A, min{S (ELEM) L(P) (VBAR) A (L-HOOK EQ) S} exists.

Each continuous extension of a partially ordered set P gives rise to a binary relation on P, called the well below relation for that continuous extension. We have characterized these relations and have shown how to construct a continuous extension having a specified well below relation. This has led to several interesting examples of continuous extensions.

The second part of my research further develops some special cases of the first part. It is mainly devoted to the study of a certain duality which exists in lattice theory, the duality between the concept of "finite" and the concept of "directed". This duality expresses itself in the existence of certain theorems of lattice theory which remain true if the words "finite" and "directed" are interchanged, for example, if in the theorem

Theorem. If L is a complete ring of sets, then the set of "finitely" algebraic elements of L is a "directedly" algebraic poset.

we interchange the concepts "finite" and "directed", we obtain the theorem:

Theorem. If L is a complete ring of sets, then the set of "directedly" algebraic elements of L is a "finitely" algebraic poset.