On ordinal sums of overlap and grouping functions on complete lattices
Introduction
Bustince et al. initially introduced the concepts of overlap functions [3], [4] and grouping functions [5] on the unit interval in 2009–2010 and 2012, respectively. As fuzzy logical operators that are not necessarily associative, overlap and grouping functions play an expressive part in applications, such as image processing [4], [30], decision making [5], [22], classification [17], [18], [20], [21], [32], [33], [34], [35], [44], fuzzy community detection problems [28] and other fields [26], [40]. Theoretically, key attributes such as migrativity, homogeneity, idempotency, cancellation law, modularity and relative distributive equations of overlap and grouping functions on the unit interval have been studied in [2], [14], [45], [48], [49], [50], [58], [59], [60], [61], [62], [63], [64]. Also, researchers considered construction methods of overlap and grouping functions, including ordinal sums [14], [50], additive generators [15], [16] and multiplicative generators [47] on the unit interval.
On the other hand, in recent years, some interesting and natural research topics for specific aggregation functions are based on more general structures than the unit interval, such as partially ordered sets and bounded lattices, including surveys involving in t-norms and t-conorms [13], [29], [57], works related to uninorms [10], [11], [12], [37], [41], studies of nullnorms [1], [7], [8], [9], [23], investigations of semi-t-operators [25], [53] and uni-nullnorms [55], [56].
As for the overlap and grouping functions, Paiva et al. introduced the notions of lattice-valued overlap and quasi-overlap functions [43] and demonstrated some properties of (quasi-)overlap functions on bounded lattices. Wang [54] proposed constructions of overlap functions on bounded lattices. In addition, Qiao [46] introduced the notions of overlap and grouping functions on complete lattices, where the continuity was replaced by meet-preserving and join-preserving properties, and gave two construction methods of them. He also proposed the notion of -overlap functions (resp. -overlap functions) on complete lattices corresponding to right continuity (resp. left continuity) on lattices by removing the join-preserving (resp. meet-preserving) property of overlap functions on complete lattices.
Constructing new functions from existing ones, ordinal sums is one of the most important construction methods in the theory of fuzzy logical operators on the unit interval and lattices. The definition of ordinal sums of t-norms was first provided in [51]. Saminger-Platz et al. [52] characterized when this ordinal sums leads to a t-norm and Ertuğrul et al. [24] modified the concept of ordinal sums. Recently, Dvořák and Holčapek [19] introduced a new ordinal sum construction of t-norms on bounded lattices based on interior and closure operators and Ouyang et al. [42] proposed an alternative definition of ordinal sums of t-norms on complete lattices. In the frame of other special aggregation functions, the concept of ordinal sums (e.g., copulas [38], [39], overlap functions [14] and grouping functions [50]) has also been explored.
Since several types of construction methods have been introduced in the literature and more and more attention has been paid to the overlap and grouping functions in the context of lattices, it seems quite natural to consider the relationship between the ordinal sum operation and the basic lattice structure in more detail, especially for overlap and grouping functions on complete lattices. To the authors' knowledge, there is still dearth of construction methods of them. Therefore, we focus our consideration on ordinal sums of overlap and grouping functions on complete lattices. These sequences are defined through their restrictions on subintervals of the lattice and we try to supplement the previous work in a new way to extend overlap and grouping functions.
Therefore, based on the Qiao's research [46], we consider an approach to obtain overlap and grouping functions on complete lattices by ordinal sums, as well as - and -overlap and grouping functions on complete lattices. To be specific, we extend the notion of ordinal sums of overlap functions from the unit interval to complete lattices directly and point out it does not necessarily yield a new overlap function. Hence, we investigate which type of complete lattices is suitable for allowing this ordinal sums to be an overlap function. Then, we give a concrete form for ordinal sums of - and -overlap functions on a more general complete lattice. Similar results of the grouping function are also presented.
The rest of this paper is organized as follows. In Section 2, we review some basic concepts needed in the sequel. In Section 3, we discuss ordinal sums of (- and -) overlap functions on complete lattices. In Section 4, we deal with ordinal sums of (- and -) grouping functions on complete lattices in a similar way. Finally, we present our conclusion.
Section snippets
Preliminaries
In this section, we recall several fundamental concepts and results about lattices and overlap and grouping functions on a complete lattice.
Ordinal sums of overlap functions on complete lattices
In this section, we deal with ordinal sums of overlap functions on complete lattices. As each of its summand carriers is described by a subinterval , we use an overlap function acting on this subinterval with the top element and the bottom element to construct ordinal sums. While the unit interval I is a chain, the case does not always hold for each complete lattice. Thus, we start with certain subintervals as summand carriers whose endpoints are comparable with all other
Ordinal sums of grouping functions on complete lattices
In this section, we focus on ordinal sums of grouping functions on complete lattices. First, we give the definition of ordinal sums of grouping functions on complete lattices directly extended from Definition 2.11. With the help of duality of overlap and grouping functions and Lemma 2.5, the proof in this section can be omitted.
Definition 4.1 Consider an arbitrary complete lattice L and a countable nonempty index set B. Let be a family of pairwise disjoint subintervals of L and a family
Concluding remarks
We investigated the ordinal sum construction for building overlap and grouping functions on complete lattices. At first, we extended the definition of ordinal sums of overlap and grouping functions on the unit interval to complete lattices and gave an example to illustrate that this does not always lead to an overlap function since it may fail to preserve joins.
Then, we proposed ordinal sums of finitely many overlap functions on a frame L that can be partitioned into a chain of subintervals and
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 article was supported by grants from the National Natural Science Foundation of China (grant nos. 11971365 and 11571010).
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