The distributive laws of fuzzy implications over overlap and grouping functions
Section snippets
A brief review on overlap and grouping functions
Overlap and grouping functions [16], [18] are introduced by Bustince et al. in 2009 and 2012, respectively. These two concepts, as specific non-associative binary aggregation functions, arise from some problems in image processing [15], classification [30] or decision making [47], where t-norms and t-conorms are commonly used [32]. But nonetheless, as a matter of fact, the associativity of t-norms and t-conorms is not forcefully needed in many cases.
In recent years, overlap and grouping
Preliminaries
In this section, we give a brief introduction to some basic concepts and related properties, which are used throughout the paper. We begin with the definition of fuzzy negations.
Definition 2.1 (See Dimuro et al. [26].) A function is said to be a fuzzy negation if the following conditions hold.
N satisfies the boundary conditions: and ; N is decreasing: if x ≤ y, then N(y) ≤ N(x).
Moreover, a fuzzy negation is called strong if it is involutive, that is,
- (3)
.
In this paper, we
The distributive laws of residual implications, (G, N)-implications and QL-operations over overlap and grouping functions
In this section, we mainly investigate the four basic distributive laws (5)–(8) of fuzzy implications over overlap and grouping functions for the residual implications derived from overlap functions, (G, N)-implications for grouping functions G and fuzzy negations N, and QL-operations derived from overlap functions, grouping functions and fuzzy negations, respectively. First, we show two lemmas and two propositions, which shall be used in the sequel.
Lemma 3.1 Let be an overlap function.
The distributive laws of general fuzzy implications over overlap and grouping functions
In this section, we propose the four basic distributive laws (5)–(8) for the general fuzzy implications over overlap and grouping functions.
Concluding remarks
In this paper, analogously to the discussions proposed in [9], [10], we investigate the four basic distributive laws of fuzzy implications over overlap and grouping functions. This work can be regarded as a state of the art survey on the related results of the distributive laws of fuzzy implications over overlap and grouping functions and the main contributions of this paper are listed as follows.
- •
We discuss the four basic distributive laws (5)–(8) for the residual implications derived from
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 paper was supported by grants from the National Natural Science Foundation of China (Grant nos. 11571010 and 61179038).
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