About: Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited     Goto   Sponge   NotDistinct   Permalink

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  • 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.
  • 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. (en)
Title
  • Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited
  • Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited (en)
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  • Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited
  • Concept lattices of isotone vs. antitone Galois connections in graded setting: mutual reducibility revisited (en)
skos:notation
  • RIV/61989592:15310/12:33143393!RIV13-GA0-15310___
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  • P(GAP202/10/0262)
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  • 15
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  • 128363
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  • RIV/61989592:15310/12:33143393
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  • formal concept analysis, fuzzy logic, Galois connections (en)
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  • NL - Nizozemsko
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  • [339BDCDD26A1]
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  • 199
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  • Bělohlávek, Radim
  • Konečný, Jan
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  • 000304221600010
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  • 0020-0255
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  • 10.1016/j.ins.2012.02.064
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  • 15310
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