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**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.

### MSC:

06B99 | Lattices |

03B52 | Fuzzy logic; logic of vagueness |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |

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\textit{R. Belohlavek} and \textit{J. Konecny}, Inf. Sci. 199, 133--137 (2012; Zbl 1248.06006)

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