By Sebastià Massanet, Joan Torrens (auth.), Michał Baczyński, Gleb Beliakov, Humberto Bustince Sola, Ana Pradera (eds.)

Fuzzy implication features are one of many major operations in fuzzy good judgment. They generalize the classical implication, which takes values within the set {0,1}, to fuzzy common sense, the place the reality values belong to the unit period [0,1]. those services usually are not simply primary for fuzzy common sense structures, fuzzy keep an eye on, approximate reasoning and specialist platforms, yet additionally they play an important position in mathematical fuzzy good judgment, in fuzzy mathematical morphology and snapshot processing, in defining fuzzy subsethood measures and in fixing fuzzy relational equations.

This quantity collects eight study papers on fuzzy implication functions.

Three articles concentrate on the development tools, on other ways of producing new sessions and at the universal houses of implications and their dependencies. articles speak about implications outlined on lattices, particularly implication capabilities in interval-valued fuzzy set theories. One paper summarizes the adequate and priceless stipulations of options for one distributivity equation of implication. the subsequent paper analyzes compositions in line with a binary operation * and discusses the dependencies among the algebraic homes of this operation and the precipitated sup-* composition. The final article discusses a few open difficulties with regards to fuzzy implications, that have both been thoroughly solved or these for which partial solutions are identified. those papers target to give today’s state of the art during this area.

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1) There exists an implication satisfying NT, EP, OP, CB, ID but not SN. Proof. The G¨odel implication 1, y, IGD (x, y) = if x ≤ y , if x > y for all x, y ∈ [0, 1]. (4) is an R-implication generated by the continuous t-norm TM (x, y) = min(x, y). Therefore IGD satisfies NT, EP, OP, CB and ID [8]. However we have for all x ∈ [0, 1]: NIGD (x) = IGD (x, 0) = 1, 0, if x = 0 . if x > 0 Thus IGD does not satisfy SN. Proposition 11. There exists an implication I satisfying NT, EP, CB, ID, CO but not SN.

1 Introduction Logical operators consist an important part in the construction of classical logic. Primary logical operators in classical logic include the negation operator ¬, the conjunction operator ∧, the disjunction operator ∨ and the implication operator →. Similarly as in classical logic, logical operators play a very important role in the framework of fuzzy logic. Corresponding to the negation operator, the conjunction Yun Shi · Bart Van Gasse · Etienne E. be ∗ Dedicated to the late Prof.

In Section 5 we will repeat these investigations but with the assumption that T2 is a nilpotent t-norm. In section 6 we discuss the system of functional equations consisting of Eq. (2) and Eq. (3) when both t-norms T1 , T2 are continuous and Archimedean. In section 7 we briefly discuss the solutions when T1 = TM minimum t-norm. In section 8 we investigate the system of functional equations consisting of Eq. (2) and Eq. (3) when T1 is a continuous t-norm and T2 is a strict t-norm. In section 9 we repeat the above investigations but with the assumption that T2 is a nilpotent t-norm.

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