Fuzzy implications are one of the two most important fuzzy logic connectives, the other being
t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued
setting.
A binary operation I on [0; 1] is called a fuzzy implication if
(i) I is decreasing in the first variable,
(ii) I is increasing in the second variable,
(iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0.
The set of all fuzzy implications defined on [0; 1] is denoted by I.
Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning,
decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates
new ways of generating fuzzy implications that are fit for a specific task. The generating methods
of fuzzy implications can be broadly categorised as in the following:
(M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL-
implications,
(M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g-
implications, or from fuzzy negations,
(M3): From existing fuzzy implications
