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Structured lattices and topological categories of \(L\)-sets. (English) Zbl 1191.03040

Summary: Goguen’s category of V-sets, whose objects are functions with values in a pre-ordered set and morphisms are suitable maps, is shown to be a topological construct if and only if the pre-ordered set is a complete lattice; in particular the category Set(L) of \(L\)-sets, also considered by Goguen, is topological over Set. The special case when the considered pre-ordered set is \(L\) with the “mapping to” relation arising from a structure \(\Phi =(\varphi _a)_{a\in L}\), where every \(\varphi _a:L\rightarrow [\perp,a]\) preserves arbitrary infs, already considered by the authors on any complete lattice \(L\), is investigated in detail.

MSC:

03E72 Theory of fuzzy sets, etc.
06B23 Complete lattices, completions
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
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