Associatively tied implications

  • Authors:

  • A. A. Abdel-Hamid;Nehad N. Morsi

  • Affiliations:

  • Department of Basic Sciences, Arab Academy for Science, Technology and Maritime Transport, P. O. Box 1029 Miami, Alexandria, Egypt;Department of Basic Sciences, Arab Academy for Science, Technology and Maritime Transport, P. O. Box 1029 Miami, Alexandria, Egypt

  • Venue:
  • Fuzzy Sets and Systems - Theme: Basic concepts
  • Year:
  • 2003

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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,gamma;))=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 TA that ties A; hence, the nomenclature. We study properties of that TA 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.