Preorders, residuation and closure spaces. (Préordres, résiduation et espaces de fermeture.) (French) Zbl 0802.06004
Summary: We construct an adjunction between the category \(\mathbb{P}\) of isotone functions between preordered sets and the category \(\mathbb{F}\) of closure spaces. We also give an adjunction between \(\mathbb{F}\) and the category \(\mathbb{L}\) of residuated functions between complete lattices. It follows that \(\mathbb{P}\) and \(\mathbb{L}\) are isomorphic to two disjoint full subcategories of \(\mathbb{F}\) whose objects we characterize.
MSC:
| 06A99 | Ordered sets |
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 06A06 | Partial orders, general |
| 54A05 | Topological spaces and generalizations (closure spaces, etc.) |
| 18B30 | Categories of topological spaces and continuous mappings (MSC2010) |
| 06B23 | Complete lattices, completions |
Keywords:
category of isotone functions between preordered sets; category of closure spaces; category of residuated functions between complete lattices; adjunctionReferences:
| [1] | Achache, A., How to fuzzify a closure space, J. Math. Anal.Appl., 130, 538-544 (1988) · Zbl 0648.06005 |
| [2] | Blyth, T. S.; Janowitz, M. F., Residuation Theory (1972), Pergamon: Pergamon Oxford · Zbl 0301.06001 |
| [3] | Mac Lane, S., Categories for the Working Mathematician (1971), Springer: Springer Berlin · Zbl 0232.18001 |
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