\({\mathbf Z}\)-continuous posets, topological aspects. (English) Zbl 0883.06007

Summary: The concept of “subset system on the category Po of posets” (Z-sets) was defined by J. B. Wright, E. G. Wagner and J. W. Thatcher [Theor. Comput. Sci. 7, 57-77 (1978; Zbl 0732.06001)]. The term Z-set becomes meaningful if we replace Z by “directed”, “chain”, “finite”. At the end of the paper [loc. cit.], the authors suggested to try to study the generalized counterpart of the term “continuous poset (lattice)” obtained by replacing directed sets with Z-sets, Z being an arbitrary subset system on Po. We present here some results concerning this investigation. If the author’s earlier results [Discrete Math. 152, No. 1-3, 33-45 (1996; Zbl 0851.06003)] are generalized counterparts of some purely order facts about continuous posets, the present paper deals with a generalized counterpart of the Scott topology on posets and some results related to this concept.


06B35 Continuous lattices and posets, applications
06A15 Galois correspondences, closure operators (in relation to ordered sets)