Soft Computing

, Volume 7, Issue 7, pp 458–467 | Cite as

Non-commutative fuzzy Galois connections

  • G. Georgescu
  • A. Popescu
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Abstract

 Fuzzy Galois connections were introduced by Bělohlávek in [4]. The structure considered there for the set of truth values is a complete residuated lattice, which places the discussion in a “commutative fuzzy world”. What we are doing in this paper is dropping down the commutativity, getting the corresponding notion of Galois connection and generalizing some results obtained by Bělohlávek in [4] and [7]. The lack of the commutative law in the structure of truth values makes it appropriate for dealing with a sentences conjunction where the order between the terms of the conjunction counts, gaining thus a temporal dimension for the statements. In this “non-commutative world”, we have not one, but two implications ([15]). As a consequence, a Galois connection will not be a pair, but a quadruple of functions, which is in fact two pairs of functions, each function being in a symmetric situation to his pair. Stating that these two pairs are compatible in some sense, we get the notion of strong L-Galois connection, a more operative and prolific notion, repairing the “damage” done by non-commutativity.

Keywords Non-commutative fuzzy logic, Fuzzy Galois connection, Fuzzy relation, Non-commutative conjunction 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • G. Georgescu
    • 1
  • A. Popescu
    • 2
  1. 1.Institute of Mathematics, Calea Grivitei Nr. 21, P.O. Box 1-767, Bucharest, Romania e-mail: georgescu@funinf.math.unibuc.roRO
  2. 2.Fundamentals of Computer Science, Faculty of Mathematics, Univeresity of Bucharest, Str. Academic Nr 14, 70109 Bucharest, Romania e-mail: uuomul@yahoo.comRO

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