Abstract

This study is an attempt to develop a theory of Z-continuous posets which generalizes Dana Scott's theory of continuous lattices. This generalization was first suggested by Wright, Wagner and Wright.

The first part of this work is devoted to finding certain natural classes of a Z-continuous mappings under which the images of Z-continuous posets are Z-continuous. The main result is the following theorem.

THEOREM: The image of a Z-continuous poset under a Z-continuous projection operator is Z-continuous.

This theorem leads to a definition of homomorphism for Z-continuous posets and also a definition of subalgebra for Z-continuous posets. We prove that homomorphic images and subalgebras of a Z-continuous poset are Z-continuous.

Next we define a basis of a Z-continuous poset. The infimum of the cardinalities of bases of a Z-continuous poset is called its weight. It is proved that the weights of homomorphic images and subalgebras of a Z-continuous poset P are less than that of P. We also study the Z-algebraic posets and generalize certain well known results about algebraic lattices to Z-algebraic posets.

When a given poset does not have certain desired properties it is natural to consider extensions of the poset which have these properties. We define the Z-continuous extensions of a poset B. We consider the set of all Z-continuous extensions of a poset B. We define a quasi-order on this set and consider the partially ordered set of equivalence classes arising from this quasi-order in the usual way. This poset is denoted by C(,Z)(B). We prove certain global and local results about C(,Z)(B). For example, we prove the following two theorems.

THEOREM: For all posets B and all union complete subset systems Z, C(,Z)(B) is a Z-complete poset with a top element.

THEOREM: Two Z-continuous extensions of a poset B are equivalent if and only if they are order isomorphic.

We also consider the Z-algebraic extensions of a poset B and show that the subposet A(,Z)(B) of C(,Z)(B) which consists of the algebraic elements is also a Z-complete poset.

Finally we study the union complete subset systems in some detail. We restrict our attention to the countable subset systems. We define two subset systems Z and Z' to be equivalent if and only if for all posets P, Z-ideals and Z'-ideals of P coincide. We analyze the structure of the poset of all equivalence classes under this equivalence relation.