In this paper, we introduce a new construction method of a fuzzy implication from n increasing functions g i : [ 0 , 1 ] → [ 0 , ∞ ) ,   ( g ( 0 ) = 0 ) ( i = 1 , 2 , … , n ,   n   ∈ ℕ ) and n + 1 fuzzy negations N i ( i = 1 , 2 , … , n + 1 ,   n   ∈ ℕ ). Imagine that there are plenty of combinations between n increasing functions g i and n + 1 fuzzy negations N i in order to produce new fuzzy implications. This method allows us to use at least two fuzzy negations N i and one increasing function g in order to generate a new fuzzy implication. Choosing the appropriate negations, we can prove that some basic properties such as the exchange principle (EP), the ordering property (OP), and the law of contraposition with respect to N are satisfied. The worth of generating new implications is valuable in the sciences such as artificial intelligence and robotics. In this paper, we have found a novel method of generating families of implications. Therefore, we would like to believe that we have added to the literature one more source from which we could choose the most appropriate implication concerning a specific application. It should be emphasized that this production is based on a generalization of an important form of Yager’s implications.

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In this article, we develop new constructed methods with specific conditions. The first method is a generalization of convex combination using n fuzzy implications. The second method is a parameterization of Lukasiewicz implication in an Ordering Property (OP) fuzzy implication form. The innovation in this work is the presentation of three new constructed methods of (OP) polynomial and (OP) rational fuzzy implications. We investigate some families of Ordering Property (OP) and Ordering Property (OP) Rational fuzzy implications. To these methods, we give some coefficient conditions in order to satisfy basic properties like ordering property (OP), identity property (IP) and contrapositive symmetry (CP). Background: Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0,1}, to fuzzy logic, where the truth values belong to the unit interval [0,1]. The study of this class of operations has been extensively developed in the literature in the last 30 years from both theoretical and applicational points of view. Introduction: In this paper, we develop five new methods for constructing fuzzy implications with specific properties. The paper starts by presenting the first fuzzy implication construction machine that uses n fuzzy implications with specific conditions. Next, we parameterize Lukasiewicz implication and create new families of (OP) polynomial and (OP) rational implications. For each method we investigate which conditions are satisfied and we give some examples. Method: The first constructed method uses n fuzzy implications in a linear product representation. The second method is an (OP) polynomial implication a parameterized Lukasiewicz implication. The third method is a rational implication with five parameters. In the fourth method we give a general form in the previous method by changing variables x and y with increasing functions. Finally, the last method is another (OP) rational implication with three parameters. Result: In each method we present the properties that are satisfied. We generalize the (OP) polynomial and rational by replacing the variables with monotonic functions or add powers on them. Finally, we generalize and we give examples of new produced fuzzy implications. Conclusion: As a future work, we can create new families of rational implications by changing the polynomials of the numerator and denominator so that they satisfy more properties. Finally, the new methods we presented can contribute in the construction of uninorms and copulas under certain conditions.