2-CONTINUOUS POSETS AND UNION COMPLETE SUBSET SYSTEMS

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.;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.;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.;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.