On homogeneous, quasi-homogeneous and pseudo-homogeneous overlap and grouping functions

Qiao, Junsheng; Hu, Bao Qing

doi:10.1016/j.fss.2018.06.001

Cited by 40 publications

(17 citation statements)

References 36 publications

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“…where A = {(x 1 , A(x 1 )), (x 2 , A(x 2 ))} and B = {(y 1 , B(y 1 )), (y 2 , B(y 2 ))} are fuzzy sets over X and Y , respectively. Then, whenever I : [0, 1] 2 → [0, 1] is an implication function, Rule (26) can be represented by means of the following matrix:…”

Section: Fuzzy Implications and The O-conditionalitymentioning

confidence: 99%

“…Similarly, the exchange principle is not required for fuzzy implication functions in several applications, e.g., in decision making [15,16,17], consensus measures [15,16], multicriteria decision problem by similarity measures [18]. See the discussions in [4,10,19,20,21] and, for a summary of related works discussing the subject, see [22,23,24,25,26].As an example, in fuzzy modeling of pairwise comparisons [27,28], in the context of decision making based on fuzzy preference relations [29,30,31,32], in general, an indifference relation is defined using a weak preference relation given by a t-norm combination, which is a measures of the the (weak) preference of one alternative over the other and vice versa [33,34]. However, a pairwise comparison refers to only two alternatives (and the result of the comparison is not an alternative itself), and so the associative property is not a strong requirement, and then, one need not use t-norms as the combination operator [35].…”

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confidence: 99%

“…Overlap and grouping functions were extensively studied in the literature. See, e.g.,[19,20,23,25,26,39,40,41].Then, in the sequence, the concept of fuzzy implications derived from overlap and grouping functions were introduced. In [21,42], based on residual implicators of general conjunctions [43,44], Dimuro et al introduced R O -implications, the residual implications derived from overlap functions O, preserving the residuation property.…”

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confidence: 99%

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confidence: 99%

“…Overlap and grouping functions were extensively studied in the literature. See, e.g.,[19,20,23,25,26,39,40,41].…”

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confidence: 99%

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The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions

Dimuro

Bedregal

Fernández

et al. 2019

International Journal of Approximate Reasoning

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Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications R O -implications, the strong implications (G, N )-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N . Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x, I(x, y)) ≤ y, for any fuzzy implication I and overlap function O, and, in particular, for R O -implications, (G, N )-implications, QL-implications and D-implications derived from tuples (O, G, N ), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T -conditionality and U -conditionality, for t-norms T and uninorms U , respectively. © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ inference processes via the generalized Modus Ponens (GMP) and Modus Tollens (GMT), carried out via the Zadeh's Compositional Rule of Inference [5]. 1 The GMP, which is the focus of this paper, can be implemented by a scheme enabled by the functional inequality called the Law of T -Conditionality, which, for a t-norm T and a fuzzy implication I, is stated by:In fact, (TC) means x * T (x → y) ≤ y, which generalizes the Modus Ponens x ∧ T (x → y) y to the fuzzy context [7, 8].In the literature, the studies on the T -Conditionality have been done just for the three main families of fuzzy implications, namely, R-, (S, N )and QL-implications. 2 R-implications are generalizations to [0, 1] of Boolean implications defined by the identity given, for a universe set X, bywhere A, B ⊆ X, and the intersection is generalized by a t-norm. This class of implications is related to a residuation concept from the intuitionistic logic. On the other hand, (S, N )-implications are generalizations to [0, 1] of the Boolean material implication defined aswhere ∨ and ¬ are...

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Section: Fuzzy Implications and The O-conditionalitymentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

“…Overlap and grouping functions were extensively studied in the literature. See, e.g.,[19,20,23,25,26,39,40,41].…”

mentioning

confidence: 99%

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The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions

Dimuro

Bedregal

Fernández

et al. 2019

International Journal of Approximate Reasoning

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ℱ‐homogeneous functions and a generalization of directional monotonicity

Santiago

Sesma‐Sara

Fernández

et al. 2022

Int J of Intelligent Sys

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A function that takes n numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor λ yields the original output multiplied by that same factor. This concept has been extended by the notion of abstract homogeneity, which generalizes the product in the expression of homogeneity by a general function g and the effect of the factor λ by an automorphism. However, the effect of parameter λ remains unchanged for all the input values. In this study, we generalize further the condition of abstract homogeneity by introducing MJX-tex-caligraphicnormalℱ‐homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs. Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions. To achieve that, we generalize directional monotonicity by MJX-tex-caligraphicnormalℱ directional monotonicity, which is defined with respect to a family of functions MJX-tex-caligraphicnormalℱ and a family of vectors MJX-tex-caligraphicscriptV. Finally, we show how the introduced concepts could be applied in two different problems of computer vision: a snow detection problem and image thresholding improvement.

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General Grouping Functions

Santos

Dimuro

Asmus

et al. 2020

Communications in Computer and Information Science

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Some aggregation functions that are not necessarily associative, namely overlap and grouping functions, have called the attention of many researchers in the recent past. This is probably due to the fact that they are a richer class of operators whenever one compares with other classes of aggregation functions, such as t-norms and t-conorms, respectively. In the present work we introduce a more general proposal for disjunctive n-ary aggregation functions entitled general grouping functions, in order to be used in problems that admit n dimensional inputs in a more flexible manner, allowing their application in different contexts. We present some new interesting results, like the characterization of that operator and also provide different construction methods.

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On homogeneous, quasi-homogeneous and pseudo-homogeneous overlap and grouping functions

Cited by 40 publications

References 36 publications

The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions

The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions

ℱ‐homogeneous functions and a generalization of directional monotonicity

General Grouping Functions

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