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**\({\mathbf Z}\)-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.