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A uniform approach to inductive posets and inductive closure. (English) Zbl 0732.06001
Summary: The definition scheme: “A poset P is Z-inductive if it has a subposet B of Z-compact elements such that for every element p of P there is a Z-set S in B such that \(p=\sqcup S\),” becomes meaningful when we replace the symbol Z by such adjectives as “directed”, “chain”, “pairwise compatible”, “singleton”, etc. Furthermore, several theorems have been proved that seem to differ only in their instantiations of Z. A similar phenomenon occurs when we consider concepts such as Z-completeness or Z- continuity. This suggests that in all these different cases we are really talking about the same thing. We show that this is indeed the case by abstracting out the essential common properties of the different instantiations of Z and proving common theorems within the resulting abstract framework.

MSC:
06A06 Partial orders, general
68Q55 Semantics in the theory of computing
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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