A remark on intuitionistic fuzzy implications

In the paper an attempt is made at introducing a classification scheme for some of the intuitionistic fuzzy implications. This has allowed us to navigate the existing implications in a more consistent manner and has revealed a duplicate implication.


Introduction
At present there are 190 proposed intuitionistic fuzzy implications defined in a series of papers [1-10, 12-17, 19-39, 41-46] most of which are collected in [11]. For alternative points of view on intuitionistic fuzzy implications we refer the interested reader to [40] and [47].
In order to make research into the existing 190 implications, we further focus our attention on the functions used to represent them (sg, sg, min, max, ·) and we propose a sample classification based on their properties. This has allowed us to identify a duplicate implication, namely → 40 which coincides with → 173 .

Preliminaries
Here we remind some of the basic definitions which will be used later. Definition 1. The functions sg and sg are defined for the real variable x as follows: Remark 1. From Definition 1 it is seen that the following equality holds: Next we recall the following: Definition 2 (cf. [18]). An intuitionistic fuzzy pair (IFP) is an ordered couple of real non-negative numbers a, b , with the additional constraint: If we denote the set of all IFPs by U IFP , we can view an implication as a particular mapping of the kind (bound by additional constraints due to the Axioms that need to be satisfied): In other words all implications are of the form: where x ∈ U IFP , y ∈ U IFP , f (x, y), g(x, y) ∈ U IFP . In our further considerations we suppose that everywhere x = a, b and y = c, d .
The list of all such implications is given below: We have marked → 40 and → 173 by * to denote the fact that they coincide. Some of the remaining implications can be represented in the form where f (x, y), g(x, y) ∈ [0, 1]. Namely, One can easily observe that (7) may be treated as a particular case of (8) with the special choice of g(x, y) = 1 ∀x, y ∈ U IFP .
However, such approach while technically correct does not yield particularly useful information. The rest of the implications have less tractable representations. However, implications that satisfy (7) and/or (8) may be studied based on the properties of the functions f (x, y) and g(x, y), which allows for a more unified approach in treating them.
In the light of the above we can formulate the following Open problem. Can implications that do not satisfy (7) or (8) be categorized in suitable classes which can be described by a single formula?

Conclusion
In the present paper we proposed a partial classification based on the representation of the existing implications. This allows not only to study implications which satisfy (7) and/or (8) in a unified manner, but also to introduce and study new implications with such property. It also helps in detecting duplicating or coinciding implications as was the case with implications → 40 and → 173 .