On the Generalized Cross-Law of Importation in Fuzzy Logic

: Recently, Baczyński et al. introduced two pexider-type generalisations of the law of importation in fuzzy logic, i.e., y (CLI), where C is a fuzzy conjunction and I , J are fuzzy implications. However, (CLI) has not been adequately investigated so far. In this paper, we firstly show that (CLI) can be derived from the α-migrativity of an R -implication obtained from an α-migrative t-norm. Secondly, the relationships between the satisfaction of the law of importation (LI) by the pairs ( C , I ) or ( C , J ) and the satisfaction of (CLI) by the triple ( C , I , J ) are studied. Moreover, some necessary conditions of (CLI) are given. Finally, we study (CLI) under three


On the Laws of Importation in Fuzzy Logic
Fuzzy logic connectives attract a good deal of attention for research because of their interesting properties and wide range of applications, not only in approximate reasoning and fuzzy control, but also in many other research area they have proved to be valuable like composition of fuzzy relations [1,2], fuzzy relational equations [3,4], fuzzy mathematical morphology [5], fuzzy neural networks [6], fuzzy rough sets [7][8][9] and data mining [10]. This fact has led more and more people to a systematic research of many fuzzy logic connectives in theory, analyze additional properties of fuzzy implications and solve functional equations involving this kind of operators (see the recent survey [11][12][13]).
In the framework of fuzzy logic, the law of importation on fuzzy implications plays an important role in fuzzy relational inference mechanisms (FRIM), since one can generate an equivalent multilayered scheme that markedly improves the computational efficiency of the whole system (see [14]). Furthermore, some applications of the law of importation dealing with Zadeh's compositional rule of inference (CRI) have been studied in [15], and most of them believed it is necessary to theoretically study this law before applying it.
In classical two-valued logic, one of the most important tautologies is the following law of importation: ( ) ( ( )). p q r p q r      (1) The general form of the above equivalence is the well-known law of important (LI, for short): ( ( , ), ) ( , ( , )), , , [0, 1], I T x α y I x I α y α x y   (2) where T is a t-norm and I is a fuzzy implication. Moreover, Mas et al. [16] extended the above equation to the following form: c ( ( , ), ) ( , ( , )), , , [0, 1], I U x α y I x I α y α x y   (3) where c U is a conjunctive uninorm and I is a fuzzy implication derived from uninorms. There are some results already known about this property. Specifically, in A-implications defined by Türkşena et al. [17], the Equation (2) with T as the product t-norm  P ( , ) T x y xy was taken as one of the axioms.
Later, Mas et al. [18] studied the law of important for (S, N)-, R-, QL-and D-implications derived from smooth discrete t-norms and t-conorms. In [19], Massanet and Torrens introduced a weaker version of Equation (2), called the weak law of important (WLI, for short): ( ( , ), ) ( , ( , )), , , [0, 1], I F x α y I x I α y α x y   (4) where F is a commutative, conjunctive, and non-decreasing function. Moreover, they have made new characterizations of (S, N)-implications, R-implications, and their counterparts for uninorms based on Equation (4). Therefore, it seems interesting and important to study various laws of importation in fuzzy logic.

Motivation of This Work
Recently, Baczyński et al. [20] have generalized Equation (2) to the following functional equations through the α-migrativity of fuzzy implications, called generalized laws of importation, as can be seen below: where I and J are fuzzy implications and C is a fuzzy conjunction. Note that when  C T and  , J I then both Equations (5) and (6) reduce to Equation (2). In other words, Equation (2) is a special case of Equations (5) and (6). However, the generalized cross-law of importation (6) has not been investigated yet. A meaningful way of establishing the connection between (6) and the law of α-migrativity is desired. Moreover, there are two questions that we want to study: Are there different implication functions I and J such that (6) holds for some fuzzy conjunction C? If the answer is yes, then the next question arises: Are there triples (C, I, J) that satisfy both (5) and (6)? Both of those questions are answered in this paper.
In this work, we want to study the recently proposed property of generalized cross-law of importation in fuzzy logic. Furthermore, we go on to investigate the relationship with α-migrativity and some conditions to satisfy the studied functional equations. On the one hand, we hope that our work give a chance for better understanding of a connection between the law of importation and the law of α-migrativity. On the other hand, we believe that the connection will be useful in results and applications wherein (LI) plays a key role (see [15,21,22] for details).

Novelties of This Work
The novelties of this work are threefold: (i) Showing the generalized cross-law of importation can be derived from the α-migrativity of an R-implication obtained from an α-migrative t-norm. (ii) Discussing the relationship between Equations (2), (5) and (6) under three different perspectives. (iii) Extending Equations (5) and (6) to a more generalized version which depend on four functions.
The structure of the paper is organized as follows: In Section 2, we recall some basic definitions and provide several examples that are useful in further considerations. In Section 3, we give some necessary conditions of (6), and study the generalized cross-law of importation from three different perspectives. In Section 4, we discuss with different results given in the previous section and present some issues worth further investigate. Finally, Section 5 covers some conclusions.

Preliminaries
In order to help the reader ger familiar with the theory, we recall here some basic definitions and facts which are necessary for the development of this article. More details about t-norms, fuzzy negations, fuzzy implications, and fuzzy conjunctions can be found in [23][24][25].

Definition 4 ([25]). A fuzzy implication I is said to satisfy
(i) The boundary property if: (16) where N is a (continuous, strict, strong) fuzzy negation.
Example 1 ([25]). Examples of fuzzy implications that are used in this paper are:

The least Lt
I and the greatest Gt I fuzzy implications: (i) I is an (S, N)-implication generated from some t-conorm S and some continuous (strict, strong) fuzzy negation N. (ii) I satisfies (12), (17) and  ( , 0) ( ) I x N x is a continuous (strict, strong) fuzzy negation.
Moreover, the representation  ( , ) ( ( ), ) I x y S N x y is unique in this case.
 (1, 1) 1. C (20) Remark 3. A fuzzy conjunction C which satisfies (7), (8) and (10) is a t-norm. Every t-norm is a fuzzy conjunction, but the converse is not true. The left-and right-neutral elements of C will be denoted by l e and , r e respectively. The family of all fuzzy conjunctions will be denoted by .  Example 2 ([20]). Here are some fuzzy conjunctions which will be used in this paper: , ,

Solutions of Equation (6)-Some Necessary Conditions
Remark 4. It can be shown that the generalized cross-law of importation (6) can be derived from the αmigrativity of an R-implication obtained from an α-migrative t-norm. Now, we shall consider the following cases: Note that as z varies over [0, 1], αz varies over [0, ] α and substituting  β αz in to (21), we obtain Note that (22) can be expressed as  C I J that satisfy (6). For example, consider the least and the greatest fuzzy implications (see .

I J   It can be shown that the satisfaction of (2) by either/both the pairs ( , )
C I and ( , ) C J is neither sufficient nor necessary for the triplet ( , , ) C I J to fulfill (6). Consider the following fuzzy implications and the result is presented in Table 1.  (2) and (6).

Perspective Two:
The Pair (C, J) Satisfies (2) In this subsection, we focus on the second perspective when the pair (C, J) satisfies (2). Then the following items hold: (1) If (a) is true, then (ii) (iv).  (1) If , l r f e  then (ii) (iv)  follows from Proposition 1 (ii).
(2) Necessity. Assume that the pair (C, J) satisfies (2). We already know that every t-norm is a fuzzy conjunction. By the commutativity of a t-norm, if an implication J satisfies (2) with respect to any tnorm T, then J satisfies (17). Sufficiency. Assume that J satisfies (17). Note that J is a fuzzy implication it satisfies (12) and then Theorem 2 implies that J is an (S, N)-implication derived from a t-conorm S and a continuous fuzzy negation N. Moreover, ,   (6) with the corresponding pair (C, J) satisfies (2). See for instance Example 3.

Perspective Three:
The Triple (C, I, J) Satisfies (5) and (6) In this subsection, we want to discuss the second question as we have mentioned in the introduction, i.e., are there exist triples ( , , ) C I J I  that satisfy both (5) and (6)?

Remark 9.
(i) We already know that the triples 0 In a similar way as in Remark 6, one can show that the satisfaction of Equation (5) by the triple ( , , ) C I J is neither sufficient nor necessary to satisfy Equation (6). The result is presented in Table 2.  (5) and (6).

C I J (C, I, J) Satisfies (5) (C, I, J) Satisfies (6)
(iii) Note that if the triples ( , , ) C I J that both satisfy (5) and (6)   On the other hand, it is easy to verify that the triple  to an α-migrative t-norm (Remark 4). Another important fact is that the satisfaction of (2) by either/both the pairs (C, I) and (C, J) is neither sufficient nor necessary for the triplet (C, I, J) to satisfy (6) (Remark 6). In addition, some necessary conditions for solutions to Equation (6) are given (Proposition 1). Following this, we have discussed the relationship between Equations (2), (5) and (6) under three different perspectives. In particular, note that both Equations (5) and (6) can be further generalized as mentioned in Remark 9 (iv). We believe that our work provides an opportunity for better understanding of a connection between the laws of importation and the laws of α-migrativity.