On ordinal sums of countably many - and -overlap functions on complete lattices
Introduction
Bustince et al. [3] presented the axiomatic definition of overlap functions to identify objects in a given image. Since then, overlap functions have been developed rapidly in application, especially in classification [19], [20], [10], [11], fuzzy community detection problems [17], image processing [3], decision making [4], [14], forest fire detection [15] and power quality [22]. Theoretically, key attributes such as (cross) migrativity [39], [30] and homogeneity [29] have been discussed. Theoretical findings also include relative distributive equations [36], [18], [38] and induced implications of overlap functions on the unit interval [7], [9]. Besides, researchers considered construction methods, including ordinal sums [6], additive generators [8] and multiplicative generators [28]. Researchers have considered relaxation of overlap functions, such as general overlap functions [21]. Recently, Zhang et al. [37] removed the right continuity of general overlap functions to obtain the so-called semi-overlap functions.
In application fields, such as decision making and experts systems, many types of data that need to be processed may contain uncertainty, which can be expressed by using intervals, fuzzy numbers and different types of fuzzy sets [2]. These fuzzy sets can be encompassed in the notion of L-fuzzy sets, i.e., fuzzy sets where the membership degrees take values in lattices. As Paiva et al. [24] have said, potential applications of overlap functions on a lattice L could be the construction of dilations and erosion operators in L-fuzzy mathematical morphological by using L-overlap functions as the conjunction operators and their residuated L-fuzzy implications in [[1] Definition 9].
Therefore, Paiva et al. [25] proposed the extension of overlap functions to the lattices in two ways, namely, removing of the continuity requirement to get the so-called quasi-overlap functions or considering Scott’s continuity instead of Euclidean continuity on the unit interval . Meanwhile, Qiao [26] replaced continuity of overlap functions on with sup- and inf-preserving properties on complete lattices. In fact, these two independent ways of definition are equivalent. Naturally, further extensions such as -overlap functions and -overlap functions [26] on complete lattices were presented. Specifically, a -overlap function (resp. -overlap function) is a commutative and increasing function which satisfies boundary conditions of overlap functions and left (resp. right) continuity on lattices. Notice that semi-overlap functions mentioned before is equivalent to -overlap functions when the underlying structure is the unit interval . Based on this, Wang and Hu [33], [34] further studied construction methods and -migrativity of (- and -) overlap functions on a complete lattice.
On the other hand, ordinal sums [5] provide a way to create, from a family of operators of a certain class, a new operator of the same class. This method was applied to triangular norms (t-norms, for short) [32], which is a special kind of semigroup on the unit interval. Subsequently, Saminger [31] discussed ordinal sums of t-norms on certain lattices. Recently, scholars [12], [13] constructed a different ordinal sum of t-norms on bounded lattices on the basis of so-called e-operators. Meanwhile, Ouyang et al. [23] partitioned a given bounded lattice and constructed ordinal sums of t-norms on a complete lattice based on this partition. Qiao also [27] investigated ordinal sums of discrete quasi-overlap functions on a finite chain.
In [34], the authors obtained an overlap function by means of ordinal sums of finitely many overlap functions by dividing the underlying frame into a family of subintervals. When there are elements incomparable with the end points of these subintervals, they investigated ordinal sums of - and -overlap functions. Based on this work, we continue to explore ordinal sums of countably many - and -overlap functions on subintervals of a complete lattice, especially for countably infinite ones.
As we have stated, there are potential applications for extensions of overlap functions on lattices in decision making and so on. Besides, Paiva et al. [24] proposed notion of residuation for quasi-overlap functions on lattices, which enables the application of overlap functions on lattices in fields like image processing or Formal Logic. In the field of logic, residuation is an essential algebraic property that is required to have good semantics for fuzzy logic systems based on the modus ponens rule. In fact, left-continuity is the necessary and sufficient condition for a fuzzy conjunction having a residuum. Using this fact, researchers proposed a new classification algorithm based on semi-overlap functions and quintuple implication principle method for fuzzy modus ponens problems [37]. This motivates us to consider the construction method of -overlap functions on complete lattices, which may bring new perspectives of studying their residual implications. Meanwhile, inspired by ordinal sums of overlap function, researchers have generalized new ways of ordinal sums for fuzzy implications. Study related to ordinal sums of - and -overlap functions on lattices may bring inspirations for ordinal sums of fuzzy implications on lattices.
The rest of this paper is structured as follows. In Section 2, we review some basic concepts needed in the sequel. In Section 3, we propose the ordinal sum of -overlap functions on a complete lattice based on its partition and prove it to be a -overlap function. In Section 4, we discuss ordinal sums of -overlap functions based on a dual partition. Examples fitting in such ordinal sums of - and -overlap functions on certain complete lattices are also given. Finally, we present our conclusion.
Section snippets
Preliminaries
In this section, we recall basic concepts and results about lattices and overlap functions on complete lattices.
Ordinal sums of countably many -overlap functions on complete lattices
In this section, we first deal with ordinal sums of -overlap functions on contiguous subintervals of certain complete lattice. Definition 3.1 Let L be a complete lattice. Consider such that and denote . Take as -overlap functions on as a -overlap function on and as a -overlap function on . The ordinal sum is given by
Ordinal sums of countably many -overlap functions on complete lattices
At first, dual to the partition of a complete lattice given in Section 2, we give another partition of , while is unchanged.
Consider with . Then we can get and . In addition, and , whereand
Besides, we denotewhere . Theorem 4.1 Let L be a complete lattice with prime and co-prime . Consider with . Let be -overlap functions on be a
Concluding remarks
This work is mainly centered on ordinal sums of countably many - and -overlap functions on subintervals of the given complete lattice, whose endpoints are comparable. The ordinal sum proposed in our work is a -overlap function (resp. -overlap function) under certain conditions, namely, being the prime and being the co-prime. This method can be applied to define the ordinal sums of - and -grouping functions on complete lattices.
As we have stated, ordinal sums can produce a new
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.
Acknowledgments
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 article was supported by grants from the National Natural Science Foundation of China (Grant Nos. 11971365 and 11571010).
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