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Associatively tied implications. (English) Zbl 1042.03021

The paper deals with algebras proper for the development of fuzzy logic. The basic algebra is a partially ordered set with a top element 1 endowed by an implication triple \((A, K, H)\) where \(A\) is an implication (antitone in the left argument, isotone in the right and has 1 as a left identity element), \(K\) is a conjunction and \(H\) is a forcing implication so that the adjointness condition holds: \[ \beta \leq A(\alpha, \gamma) \text{ iff } K(\alpha, \beta)\leq \gamma \text{ iff }\alpha \leq H(\beta, \gamma). \] An implication operator \(A\) on a complete lattice \(L\) is associatively tied if there is a binary operation \(T\) on \(L\) that ties \(A\), i.e. the identity \(A(\alpha, A(\beta, \gamma))= A(T(\alpha, \beta), \gamma)\) holds for all \(\alpha, \beta, \gamma\in L\). The authors show that there exists a binary operation \(T_A\) that ties \(A\). Properties of this operation are studied. Characterizations for the validity of associative tiedness for an implication \(A\) are sought.

MSC:

03B52 Fuzzy logic; logic of vagueness
03G25 Other algebras related to logic
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