Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited. (English) Zbl 1248.06006

Summary: It is well known that concept lattices of isotone and antitone Galois connections induced by an ordinary binary relation and its complement are isomorphic, via a natural isomorphism mapping extents to themselves and intents to their complements. It is also known that in a fuzzy setting, this and similar kinds of reduction fail to hold. In this note, we show that when the usual notion of a complement, based on a residuum w.r.t. 0, is replaced by a new one, based on residua w.r.t. arbitrary truth degrees, the above-mentioned reduction remains valid. For ordinary relations, the new and the usual complement coincide. The result we present reveals a new, deeper root of the reduction: It is not the availability of the law of double negation but rather the fact that negations are implicitly present in the construction of concept lattices of isotone Galois connections.


06B99 Lattices
03B52 Fuzzy logic; logic of vagueness
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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