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Two types of implications derived from uninorms. (English) Zbl 1125.03018
A uninorm is a function \(U: [0, 1]\times [0, 1]\to [0, 1]\) which is associative, commutative, increasing in both variables and such that there exists an element \(e\in [0, 1]\), called the neutral element, such that \(U(e, x)= x\) for all \(x\in [0, 1]\). \(U\) becomes a t-norm if \(e=1\) and a t-conorm if \(e=0\). Thus, the theory of uninorms includes the theory of t-norms and t-conorms. A uninorm is conjunctive if \(U(1, 0)=0\) and disjunctive if \(U(1, 0)=1\).
Thus, a natural question is raised, how implications can be introduced on the basis of uninorms. There are essentially two ways: QL-implications \(I_Q(x, y)= U'(N(x), U(x, y))\), where \(U\) is a conjunctive and \(U'\) a disjunctive uninorm and \(N\) is a strong negation. The second implication is D-implication defined by \(I_D(x, y)= U'(U(N(x), N(y)), y)\). The paper contains several technical results concerning the behavior of both kinds of implications in the continuous as well as in the non-continuous case, including necessary and necessary and sufficient conditions for their existence.

MSC:
03B52 Fuzzy logic; logic of vagueness
68T37 Reasoning under uncertainty in the context of artificial intelligence
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