Non-commutative fuzzy Galois connections

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.

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Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday.

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Georgescu, G., Popescu, A. Non-commutative fuzzy Galois connections. Soft Computing 7, 458–467 (2003). https://doi.org/10.1007/s00500-003-0280-4

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