Associatively tied implications

doi:10.1016/S0165-0114(02)00268-3

Fuzzy Sets and Systems

Volume 136, Issue 3, 16 June 2003, Pages 291-311

https://doi.org/10.1016/S0165-0114(02)00268-3 Get rights and content

Abstract

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(α,A(β,γ))=A(T(α,β),γ) holds for all α, β, γ in L. This property extends to multiple-valued logic the following equivalence in classical logic: $(X⇒(Y⇒Z))≡((X&Y)⇒Z)$ . 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.

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Associatively tied implications

Abstract

Part I: Inform. Sci.

Part II: J. Fuzzy Math.

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Internat. J. Man–Machine Stud.

Fuzzy Sets and Systems

Basic properties of implicators in a residual framework

Tatra Mt. Math. Publ.

Abstract residuation over lattices

Bull. Amer. Math. Soc.

Residuated lattices

Trans. Amer. Math. Soc.

Fuzzy logics and the generalized modus ponens revisited

Internat. J. Cybernet. Systems