On fuzzy implication operators
Fuzzy Sets and Systems
Fuzzy sets and fuzzy logic: theory and applications
Fuzzy sets and fuzzy logic: theory and applications
Contrapositive symmetry of fuzzy implications
Fuzzy Sets and Systems
Triangular norms on product lattices
Fuzzy Sets and Systems - Special issue on triangular norms
R0 implication: characteristics and applications
Fuzzy Sets and Systems - Mathematics
Automorphisms, negations and implication operators
Fuzzy Sets and Systems - Implication operators
On contra-symmetry and MPT conditionality in fuzzy logic
International Journal of Intelligent Systems
On aggregation operators for ordinal qualitative information
IEEE Transactions on Fuzzy Systems
Smooth associative operations on finite ordinal scales
IEEE Transactions on Fuzzy Systems
On the representation of fuzzy rules
International Journal of Approximate Reasoning
Modus ponens and modus tollens in discrete implications
International Journal of Approximate Reasoning
The law of importation for discrete implications
Information Sciences: an International Journal
Finite-valued indistinguishability operators
International Journal of Approximate Reasoning
Residual implications on the set of discrete fuzzy numbers
Information Sciences: an International Journal
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This paper deals with two kinds of implications defined from t-norms, t-conorms and strong negations on a finite chain L: 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). They are called QL-implications and NQL-implications respectively. We mainly study those QL- and NQL-implications derived from smooth t-norms and smooth t-conorms. It is characterized when functions defined in these ways are implication functions, and their analytical expressions are given. It is proved that both kinds of implications agree. Some additional properties are studied like contrapositive symmetry, the exchange principle and others. In particular, it is proved that contrapositive symmetry holds if and only if S is the only Archimedean t-conorm on L, and T jointly with its N-dual t-conorm satisfy the Frank equation. Finally, some QL- and NQL-implications are also derived from non-smooth t-norms or non-smooth t-conorms and many examples are given showing that in this non-smooth case, QL- and NQL-implications remain strongly connected.