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Conjunctors and their Residual Implicators: Characterizations and Construction Methods

Conjunctors and their Residual Implicators: Characterizations and Construction Methods,10.1007/s00009-007-0122-1,Mediterranean Journal of Mathematics,

Conjunctors and their Residual Implicators: Characterizations and Construction Methods   (Citations: 19)
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.  In many practical applications of fuzzy logic it seems clear that one needs more flexibility in the choice of the conjunction: in particular, the associativity and the commutativity of a conjunction may be removed. Motivated by these considerations, we present several classes of conjunctors, i.e. binary operations on [0, 1] that are used to extend the boolean conjunction from {0, 1} to [0, 1], and characterize their respective residual implicators. We establish hence a one-to-one correspondence between construction methods for conjunctors and construction methods for residual implicators. Moreover, we introduce some construction methods directly in the class of residual implicators, and, by using a deresiduation procedure, we obtain new conjunctors.
Journal: Mediterranean Journal of Mathematics - MEDITERR J MATH , vol. 4, no. 3, pp. 343-356, 2007
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    • ...Not only t-norms and t-conorms, but also copulas, quasi-copulas and even conjunctors in general ([9]), representable aggregation functions ([10]), and mainly uninorms ([11], [12], [13], [14], [15], [16])...

    Sebastia Massanetet al. Implications generated from additive generators of representable unino...

    • ...in [14 ]o r [19], where many interesting results can be found...

    Michal Boturet al. Commutative basic algebras and non-associative fuzzy logics

    • ...CI is also known as the deresiduum of I (see e.g., [20])...
    • ...Remark 7.4. From the above result and Definition 4.4, it is clear that the class of monotonic duality fitting implicators are exactly the class of fuzzy implications that satisfy (LNP) and (OP). Definition 7.5 (cf.[37,20])...
    • ...Theorem 7.8. ([20], Theorem 3.2 and Proposition 3.3)...
    • ...Now, we give a characterisation of monotonic left-continuous duality fitting conjuctors C, i.e., left-continuous commutative semi-copula, such that their residuals IC are not only duality fitting implicators but also such that every C-fuzzy equivalence relation E on a set X is also an IC-fuzzy equivalence relation and vice-versa, i.e., E 2 CFECðX Þ( ) E 2 IFEIC ðXÞ. Theorem 7.9. ([20], Theorem 3.5)...

    Balasubramaniam Jayaramet al. I-Fuzzy equivalence relations and I-fuzzy partitions

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