Abstract
AbstractWe revisit the problem of Stone duality for lattices with quasioperators, presenting a fresh duality result. The new result is an improvement over that of our previous work in two important respects. First, the axiomatization of frames is now simplified, partly by incorporating Gehrke’s proposal of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt’s recently proposed definition of bounded morphisms for polarities. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be regarded as fragments of sorted residuated (poly)modal logics, a research direction recently initiated by this author.
Funder
University of Thessaly Central Library
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference38 articles.
1. Allwein, G., Hartonas, C.: Duality for bounded lattices. Tech. Rep. IULG-93-25, Indiana University Logic Group (1993)
2. Birkhoff, G.: Lattice theory, third edn. American Mathematical Society Colloquium Publications 25, American Mathematical Society, Providence, Rhode Island (1979)
3. Craig, A.: Canonical extensions of bounded lattices and natural duality for default bilattices. Ph.D. thesis, University of Oxford (2012)
4. Craig, A., Gouveia, M.J., Haviar, M.: TiRS graphs and TiRS frames: a new setting for duals of canonical extensions. Algebra Universalis 74, 123–138 (2015)
5. Craig, A., Haviar, M.: Reconciliation of approaches to the construction of canonical extensions of bounded lattices. Math. Slovaca 64, 1335–1356 (2014)
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