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Galois connections for bilattices. (English) Zbl 07364297

Summary: We introduce notions of Galois biconnections, intended to be the bilattice analogue of classical Galois connections between lattices. A bidirectional Galois biconnection is a (compatible) pair of Galois connections, the first relating truth orderings and the second relating knowledge orderings, while a unidirectional Galois biconnection is a Galois connection equipped with extra properties that seek to capture the bilattice structure. A further distinction is made between strong Galois biconnections which furnish bilattice-isomorphic images and regular Galois biconnections which induce order-isomorphic images of the maps. We investigate all four species of Galois biconnections on pre-bilattices and on bilattices with negation and conflation. We examine both the survival of elegant properties of Galois connections (composability, invertibility, preservation of joins and meets, etc.) and the preservation of interesting bilattice properties (distributivity, boundedness, interlacing) for the images of the bilattices under the Galois biconnection. Finally, we discuss the naturally emerging biclosure operators on bilattices.

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B05 Structure theory of lattices
03G25 Other algebras related to logic
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