## $${\mathbf Z}$$-continuous posets.(English)Zbl 0851.06003

Summary: The concept of subset system on the category $${\mathbf P} {\mathbf o}$$ of posets $$({\mathbf Z}$$-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 $${\mathbf Z}$$-set becomes meaningful if we replace $${\mathbf Z}$$ by ‘directed’, ‘chain’, ‘finite’. At the end of that paper [loc. cit.], the authors suggested an attempt to study the generalized counterpart of the term ‘continuous poset (lattice)’ obtained by replacing directed sets by $${\mathbf Z}$$-sets, $${\mathbf Z}$$ being an arbitrary subset system on $${\mathbf P} {\mathbf o}$$. We present here some results concerning this investigation. These results are generalized counterparts of some purely order-theoretical facts about continuous posets.

### MSC:

 06B35 Continuous lattices and posets, applications 06A15 Galois correspondences, closure operators (in relation to ordered sets) 68Q55 Semantics in the theory of computing

Zbl 0732.06001
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### References:

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