On the ordinal sum of fuzzy implications: New results and the distributivity over a class of overlap and grouping functions
Introduction
Overlap functions [10], [11] were firstly proposed by Bustince et al. in 2009–2010, which originated from classification problems to measure the overlapping degree between the functions representing the object and background in image processing. As a complement to the notion of overlap functions, grouping functions were introduced by Bustince et al. [12] in 2012, which combined a degree of comparability between two alternatives. Furthermore, it should be pointed out that, the grouping function could be obtained by the duality with respect to the standard fuzzy negation from an overlap function, and vice-versa [9].
Overlap and grouping functions have been widely used in image processing [11], [31], decision making [12], [27] and classification [22], [23], [28], [34], [35], [36], [37], [42], which were mostly considered to use t-norms and t-conorms earlier [29]. Actually, in many situations, the associativity of t-norms and t-conorms is not necessary, which is why overlap and grouping functions were originally defined. Because of their advantages in practical applications, overlap and grouping functions also have developed rapidly in theory. Many scholars are committed to in-depth theoretical studies of them, such as some properties (e.g., migrativity, homogeneity, idempotency, cancellation, limiting, modularity condition) [9], [16], [44], [45], [55], the additive (multiplicative) generator pairs [18], [46] and the related extension studies [1], [8], [13], [15], [30], [43], [47], [50].
Meanwhile, overlap and grouping functions are another extension of the intersection and union operations on the unit interval that differ from t-norms and t-conorms, respectively, although they are closely related to continuous t-norms and t-conorms without divisors of zero. The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, involving some properties, such as idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap (grouping) functions, as discussed in [17], [20], [21]. In order to more clearly show the relationship between t-norms and overlap functions, as well as the relationship between t-conorms and grouping functions, Fig. 1 is given.
Moreover, fuzzy implications [4], [19], [38], [57] are the natural generalization, in many ways, to the unit interval, of the different equalities defining the classical logic implication connective, which were applied in approximate reasoning and fuzzy control theory. On the one hand, a focus of investigations on fuzzy implications is the distributivity equations related to them. In the following, there are the four basic distributive equations involving an implication, each of which is a tautology in classical logic: As the generalizations of Eqs. (1)–(4), the distributivity equations of fuzzy implications over overlap and grouping functions are given below: where I is a fuzzy implication, are overlap functions, and are grouping functions. The above equations show great potential in solving the problem of rule explosion hindering the development of fuzzy systems [5], [6], [51], [56].
On the other hand, various ways to generate fuzzy implications have been continuously investigated in many recent works, one of which is considering the ordinal sum of fuzzy implications. After Su et al. [52] introduced an ordinal sum of fuzzy implications by considering the structure of the ordinal sum of t-norms in 2015, many other sorts of ordinal sums of fuzzy implications have been studied (see, e.g., [2], [3], [14], [25], [26], [39], [40], [41], [53], [58]). Particularly, Baczyński et al. [2] investigated two new sorts of ordinal sums of fuzzy implications, inspired by the ordinal sum construction of overlap functions [16]. More precisely, the fuzzy implication generated by the first method (see [2, Definition 15]) is our goal to study in this paper, and we choose it mainly because of the following three aspects:
- (1)
No any additional assumptions on summands. It means that we could discuss about different intervals and fuzzy implications more extensively, without being limited to various conditions.
- (2)
The regions where linear transformations of given fuzzy implications are defined are both subsquares and along with the diagonal of , so that the solutions of distributivity equations of such fuzzy implication would be more concise and regular.
- (3)
Whether this ordinal sum of fuzzy implications fulfills the left-neutral principle (NP) depends on whether each given fuzzy implication fulfills (NP). In this sense, we classify the set of indexes according to whether fulfills (NP), and thus discuss the solutions of Eqs. (5)–(8) through classification in turn, so as to show the difference in structure between this ordinal sum of fuzzy implications and other ones.
Recently, there are researches on the distributivity equations (5)–(8) of fuzzy implications over overlap and grouping functions. In 2018, Qiao and Hu [48] investigated Eqs. (5)–(8) for -implications, -implications and QL-operations derived from tuples , and then extended to the general fuzzy implications fulfilling some certain algebraic properties. Simultaneously, Qiao and Hu [49] characterized the binary function I that satisfies Eqs. (5) and (7) for additively generated overlap and grouping functions. Soon after, Liu and Zhao [32] gave the solutions to Eq. (7) over multiplicatively generated overlap functions.
However, in this paper, we study four distributivity equations (5)–(8) under the ordinal sum of fuzzy implications with the first method, proposed in [2], over a class of overlap and grouping functions. More precisely, we conclude that the overlap functions or grouping functions over which such ordinal sum of fuzzy implications distributes are the ordinal sum representations. It should be pointed out that, our discussions on the distributivity of this ordinal sum of fuzzy implications over overlap and grouping are different from the existing research on that over t-norms and t-conorms studied by Lu and Zhao [33]. On the one hand, as we stated before, overlap and grouping function are two specific cases of unnecessarily associative aggregation functions, although they are closely related to positive and continuous t-norms and t-conorms, respectively. On the other hand, according to the structural characteristics of such ordinal sum of fuzzy implications, we classify the set of indexes according to whether fulfills (NP), so as to study each distributivity equation more deeply and clearly with the least additional conditions. The above reasons motivate us to investigate four distributivity equations (5)–(8) of such ordinal sum of fuzzy implications over a class of overlap and grouping functions.
In addition, as a theoretical supplement to the study of the ordinal sum of fuzzy implications, we focus on studying some basic properties and showing its relations with other known classes of fuzzy implications derived from overlap and grouping functions, including the -, -, QL- and D-implications [17], [19], [20], [21], [24].
The contents are organized as follows. In Section 2, some basic concepts and results are reviewed. In Section 3, some new results on the ordinal sum of fuzzy implications with the first method are investigated, including basic properties and its relations with other known classes of fuzzy implications derived from overlap and grouping functions. In Section 4, we discuss the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. Conclusions are given in Section 5.
Section snippets
Preliminaries
In this section, we briefly recapitulate some elementary concepts and essential results, which are used in the sequel.
Definition 2.1 ([7]) A fuzzy negation is a decreasing function such that and .
Example 2.1 The least and the greatest fuzzy negation and are presented, respectively, by
Definition 2.2 ([10], [11]) A function is an overlap function if it fulfills the following conditions: O is commutative; iff ; iff ; O is
Basic properties
In this subsection, we discuss three basic properties of , e.g., iterative Boolean law, right ordering property and strong boundary condition.
Proposition 3.1 Let be a fuzzy implication defined in Definition 2.8. Then fulfills (IB) iff fulfills (IB) for all .
Proof Fix arbitrarily . For any , we have that We verify that fulfills (IB) by considering the following three cases:
Distributivity of the ordinal sum of fuzzy implications over a class of overlap and grouping functions
In this section, we study four distributivity equations (5)–(8) of over a class of overlap and grouping functions. First, we give the following six lemmas, which shall be used in the sequel.
Lemma 4.1 Let be an overlap function. Then the following statements are equivalent. O has 1 as neutral element. has 1 as neutral element for all . for any , where is defined by Eq. (10).
Proof (i)(ii): We consider the following two cases to prove that has 1 as neutral element
Conclusions
This paper mainly shows a study of the ordinal sum of fuzzy implications with the first method in [2]. The main results of this paper are listed in what follows.
- •
We investigate some basic properties of the ordinal sum of fuzzy implications, such as iterative Boolean law, right ordering property and strong boundary condition. Moreover, we obtain characterizations of such ordinal sum of fuzzy implications that is a QL-implication constructed from tuples or a D-implication derived from
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to express their sincere thanks to the Editors and anonymous reviewers for their most valuable comments and suggestions in improving this paper greatly. The work described in this paper was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11971365 and 11571010).
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