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QL-implications versus D-implications. (English) Zbl 1249.03026
Summary: This paper deals with two kinds of fuzzy implications: QL and Dishkant implications. That is, those defined through the expressions \(I(x,y)=S(N(x),T(x,y))\) and \(I(x,y)= S(T(N(x),N(y)),y)\), respectively, where \(T\) is a t-norm, \(S\) is a t-conorm and \(N\) is a strong negation. Special attention is paid to the relation between both kinds of implications. In the continuous case, the study of these implications is focused on some of their properties (mainly the contrapositive symmetry and the exchange principle). Finally, the case of non-continuous t-norms or non-continuous t-conorms is studied, deriving new implications of both kinds and showing that they remain strongly connected.

MSC:
03B52 Fuzzy logic; logic of vagueness
39B05 General theory of functional equations and inequalities
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
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References:
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