Conjunctors and their Residual Implicators: Characterizations and Construction Methods

Abstract

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.

Dependencies between fuzzy conjunctions and implications

“…Thus, there exists z < y 0 such that C (1, y 0 ) z. From the residuation principle one receives I C (1, z) y 0 > z and this is the contradiction to the assumption (6). Hence,…”

“…However, some authors (e.g. I. Batyrshin and O. Kaynak [3], F. Durante et al [6]) underline that the assumptions made on these multivalued connectives are sometimes too strong and difficult to obtain. Thus, some of the conditions are omitted.…”

A concept of duality for multivariate exchangeable survival models

“…In particular, as pointed out e.g. in [8], the term t-seminorm was already used in [15] for an equivalent notion. The ageing functions B, introduced in our specific context, are permutation-invariant, but not necessarily associative; when B is also associative then it is a triangular norm or, briefly, a t-norm (see e.g.…”

Constructions of Implications Satisfying the Order Property on a Complete Lattice

“…Based on these logical operators on [0,1] , the three fundamental classes of fuzzy implications on [0,1] , i.e., R-, S-, and QL-implications on [0,1] , were defined and extensively studied (see [4][5][6][7][8]). But, as was pointed out by Fodor and Keresztfalvi [9], sometimes there is no need of the commutativity or associativity for the connectives "and" and "or".…”