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Associatively tied implications
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DOI：10.1016/S0165-0114(02)00268-3.png)
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Abstract
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We say that an implication operator A, on a complete lattice L, is associatively tied if there is a binary operation T on L that ties A; that is, the identity A(alpha,A(beta, gamma)) =A(T(alpha, beta), gamma) holds for all alpha, beta, gamma in L. This property extends to multiple-valued logic the following equivalence in classical logic: (X double right arrow (Y double right arrow Z)) = ((XY) double right arrow Z). We show that in this case there exists an associative binary operation T-A that ties A; hence, the nomenclature. We study properties of that T-A when A is associatively tied. We then seek a characterization for the validity of associative tiedness for an implication A, phrased in terms of A and two adjoints of it. (C) 2002 Elsevier Science B.V. All rights reserved.
Keywords:
nonclassical logics
fuzzy connectives
adjointness
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