Z-continuous posets. (English) Zbl 0614.06007

In [Lect. Notes Comput. Sci. 53, 192-212 (1977; Zbl 0372.06002)] J. B. Wright, E. G. Wagner and J. W. Thatcher suggested a way to generalize Dana Scott’s continuous lattices. There the authors introduced the concept of a Z-subset system.
This paper is an attempt to develop a theory of Z-continuous posets. Section 2 is mainly devoted to finding some natural classes of Z- continuous mappings under which the images of Z-continuous posets are Z- continuous. In this section the main result is theorem 2.15 which states that the image of a Z-continuous poset under a Z-continuous projection operator is Z-continuous. From this, as a corollary, the author gets the main result of H. J. Bandelt and M. ErnĂ© [J. Pure Appl. Algebra 30, 219-226 (1983; Zbl 0523.06001)]. In Section 3 he defines a basis for a Z-continuous poset and generalizes some results about the bases of continuous lattices. In this section he also studies the Z- algebraic posets. In the last section he considers the set of all Z- continuous extensions of an arbitrary poset B and defines a quasi-order on this set finally considering the poset arising from this quasi-ordered set in the usual way. He proves certain local and global results about this poset and studies the algebraic extensions of an arbitrary poset.
Reviewer: R.A.Alo


06B35 Continuous lattices and posets, applications
06A06 Partial orders, general