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Birkhoff variety theorem and fuzzy logic

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Abstract.

An algebra with fuzzy equality is a set with operations on it that is equipped with similarity ≈, i.e. a fuzzy equivalence relation, such that each operation f is compatible with ≈. Described verbally, compatibility says that each f yields similar results if applied to pairwise similar arguments. On the one hand, algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic. On the other hand, they are the formal counterpart to the intuitive idea of having functions that are not allowed to map similar objects to dissimilar ones. In this paper, we present a generalization of the well-known Birkhoff’s variety theorem: a class of algebras with fuzzy equality is the class of all models of a fuzzy set of identities iff it is closed under suitably defined morphisms, substructures, and direct products.

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Authors and Affiliations

  1. Dept. Computer Science, Palacký University, Tomkova 40, 779 00, Olomouc, Czech Republic

    Radim Bělohlávek

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  1. Radim Bělohlávek
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Correspondence to Radim Bělohlávek.

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and Institute for Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic

Mathematics Subject Classification (2000): 03B52, 08B05

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Bělohlávek, R. Birkhoff variety theorem and fuzzy logic. Arch. Math. Logic 42, 781–790 (2003). https://doi.org/10.1007/s00153-003-0188-9

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  • DOI: https://doi.org/10.1007/s00153-003-0188-9

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