Elsevier

Fuzzy Sets and Systems

Volume 427, 20 January 2022, Pages 71-95
Fuzzy Sets and Systems

Constructing overlap and grouping functions on complete lattices by means of complete homomorphisms

https://doi.org/10.1016/j.fss.2021.03.015Get rights and content

Abstract

In this paper, construction methods of overlap and grouping functions on complete lattices via complete homomorphisms and complete 0L,1L-endomorphisms are investigated. At first, we propose the notion of O-generator triple of overlap functions. Then, properties such as (α,B,C)-migrativity, (B,C)-homogeneity, idempotency and cancellation law for the overlap functions obtained by such generator triples are discussed. Finally, we give an analogous discussion on grouping functions on complete lattices.

Introduction

Originated from problems in image processing and classification, overlap and grouping functions are two kinds of not necessarily associative fuzzy logical operators. The concepts of overlap functions [9], [10] and grouping functions [11] on the unit interval were introduced by Bustince et al. in 2009–2010 and 2012, respectively. As already stated, they are specifically important in many problems related to the fusion of information, such as image processing [10], [34], decision making [11], [29], classification [25], [26], [28], [36], [37], [38], [43], fuzzy community detection problems [33] and other fields [31], [41].

From the theoretic point of view, one of the main topics is directed towards the characterization of overlap and grouping functions on the unit interval with crucial properties, such as migrativity [8], [46], [48], [64], Archimedean property [18], modularity condition [57], homogeneity [49], relevant distributive equations [35], [47], [51], [62], [63] and so forth [5], [50]. Apart from this, interval overlap and grouping functions [2], [3], [4], [12], [44] is a hot issue as well. In addition, fuzzy implications derived from overlap and grouping functions such as (G,N)-implications [24], [55], RO-implications [19], QL-implications [21] and other fuzzy implications [12], [23], [27], [56] have been investigated.

Researchers have attached importance to aggregation functions [6], which perform the operation of combining and merging several values into a single one. In recent years, due to the close connection between fuzzy set theory and order theory, special attention has been given to the study of aggregation functions on more general structures (e.g., bounded posets and lattices). Associative aggregation functions on bounded lattices, such as t-norms and t-conorms [16], [17], [54], [61], uninorms and nullnorms [1], [13], [14], [15], [40], semi-t-operators [30], [53], uni-nullnorms and null-uninorms [58], [60] have been studied in depth, including construction methods of them. For instance, t-norms and uninorms are often constructed by unary functions called generators. Quite recently, Sun and Liu [54] extended the additive generator theorem of t-norms to the lattices.

Different from associative aggregation functions, overlap and grouping functions may be constructed by a pair of generators [20], [22], [45]. For example, in the interval-valued context, Qiao and Hu [44] and Bedregal et al. [4] presented almost simultaneously the concept of interval-valued overlap functions. The first work also introduced the concept of interval additive generators pair.

As for overlap and grouping functions on posets, Paiva et al. introduced the notions of lattice-valued overlap and quasi-overlap functions in [42] and Wang [59] proposed constructions of overlap functions on bounded lattices. Besides, Qiao [52] gave the concrete forms of the continuity of overlap and grouping functions on complete lattices, where the continuity was replaced by meet-preserving and join-preserving. He gave two construction methods of them and extended some properties, such as (α,B,C)-migrativity and (B,C)-homogeneity, where α belongs to the complete lattice and B and C are two binary operators on the complete lattice.

As we have stated, overlap and grouping functions have experienced a rapid development and play crucial parts in many application areas. For one thing, overlap and grouping functions have good qualities of mathematical basis with its continuity. They are a specific class of binary aggregation functions which are different from the common binary aggregation function t-norms and t-conorms for their boundary conditions and removal of associativity. For another, fuzzy logics are usually based on a bounded lattice truth values. For instance, being special complete lattices, finite chains are often used as input value sets in some real problems, such as image processing and uncertainty in expert systems. Thus, it deserves our further attention to study overlap and grouping functions on complete lattices.

All up, on the basis of work in [52], in order to explore their potentialities in different scenarios, we try to develop deeper study of overlap and grouping functions on complete lattices, including construction methods and their properties. To be more specific, we extend overlap functions on complete lattices from given ones based on the simultaneous use of two complete lattice homomorphisms, named O-generator triples and investigate some properties for the overlap functions obtained by such generator triples. Dually, we propose a second construction method of overlap functions from a given grouping function based on the use of complete lattice homomorphisms. An analogous discussion goes for grouping functions on complete lattices.

The rest of this paper is structured as follows. In Section 2, we recall some basic notions needed in the sequel. In Section 3, we present the construction method of overlap functions by means of complete homomorphisms between complete lattices and some related results. In Section 4, we discuss some essential properties of this kind of overlap functions on complete lattices. In Section 5, the notion of overlap functions constructing by dual complete lattice homomorphisms is proposed. In Section 6, we tackle grouping functions generated by such operators on complete lattices in a similar way. Finally, our work is summarized.

Section snippets

Preliminaries

In this section, we briefly review several basic definitions and properties on complete lattices and overlap and grouping functions on complete lattices which will be used in the sequel. In the following, the symbol Λ always denotes a nonempty index set.

The O-generated overlap functions on complete lattices

In this section, firstly, we introduce the concept of O-generator triple of an overlap function on complete lattices, which gives a way to construct overlap functions. And then, we discuss under which conditions the two complete homomorphisms and the overlap function become the generator triple of an overlap function and show some related results. Finally, we give another way to obtain an overlap function by complete 0L,1L-endomorphism.

We always suppose that L and M are complete lattices in the

Some properties on O-generated overlap functions

In this section, we discuss some properties on overlap functions generated by overlap functions and complete homomorphisms, i.e., their O-generator triples, such as the (α,B,C)-migrativity, the (B,C)-homogeneity, the idempotent and cancellation law.

The G-generated overlap functions on complete lattices

In this section, we discuss overlap functions on complete lattices generated by dual complete homomorphisms and a given grouping function in a similar way as that of Section 3.

Definition 5.1

Let θ:LM and ϑ:ML be two dual complete homomorphisms between L and M and G a grouping function on M. If the two-place function Oθ,ϑ,G:L×LL given, for all x,yL, byOθ,ϑ,G(x,y)=ϑ(G(θ(x),θ(y))) is an overlap function on L, then (θ,ϑ,G) is called a G-generator triple of the overlap function Oθ,ϑ,G and Oθ,ϑ,G is said to be

Generators of grouping functions on complete lattices

In this section, we propose construction methods of grouping functions on complete lattices by means of dual complete homomorphisms. Several related results are also showed. With the help of duality of overlap and grouping functions, proof of some theorems and propositions in this section can be omitted.

Concluding remarks

In this paper, we mainly proposed the notions of O- and G-generator triples of overlap and grouping functions, with a given overlap or grouping function and two (dual) complete homomorphisms, as a new construction method of overlap and grouping functions on complete lattices. Besides, we provided another way to extend a given overlap function or grouping function by means of complete 0L,1L-endomorphisms. We also analyzed some properties on overlap and grouping functions generated by their O

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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