## 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