Residual operators of uninorms

doi:10.1007/S005000050057

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Residual operators of uninorms

Journal Article•DOI•

Residual operators of uninorms

Bernard De Baets¹, János Fodor²•Institutions (2)

Ghent University¹, University of Veterinary Science²

23 Sep 1999-Vol. 3, Iss: 2, pp 89-100

TL;DR: Two important classes of uninorms are characterized, corresponding to the use of the minimum operator (the class Umin) and maximum operator ( the class Umax) as mean operator, and the block structure of the residual implicator and residual coimplicator of members of the classUmax is investigated.

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Abstract: Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. A uninorm shows a typical block structure and is built from a t-norm, a t-conorm and a mean operator. Two important classes of uninorms are characterized, corresponding to the use of the minimum operator (the class U min) and maximum operator (the class U max) as mean operator. The characterization of representable uninorms, i.e. uninorms with an additive generator, and of left-continuous and right-continuous idempotent uninorms is recalled. Two residual operators are associated with a uninorm and it is characterized when they yield an implicator and coimplicator. The block structure of the residual implicator of members of the class U min and of the residual coimplicator of members of the class U max is investigated. Explicit expressions for the residual implicator and residual coimplicator of representable uninorms and of certain left-continuous or right-continuous idempotent uninorms are given. Additional properties such as contrapositivity are discussed.

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Journal Article•DOI•

A Survey on Fuzzy Implication Functions

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Margarita Mas, Miquel Monserrat, Joan Torrens, Enric Trillas

01 Dec 2007-IEEE Transactions on Fuzzy Systems

TL;DR: This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators.

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Abstract: One of the key operations in fuzzy logic and approximate reasoning is the fuzzy implication, which is usually performed by a binary operator, called an implication function or, simply, an implication. Many fuzzy rule based systems do their inference processes through these operators that also take charge of the propagation of uncertainty in fuzzy reasonings. Moreover, they have proved to be useful also in other fields like composition of fuzzy relations, fuzzy relational equations, fuzzy mathematical morphology, and image processing. This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators. The four most usual ways to define these implications are recalled and their characteristic properties stated, not only in the case of [0,1] but also in the discrete case.

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353 citations

Cites background or methods from "Residual operators of uninorms"

...Works including the study of implications derived from uninorms are, among others, [14], [59], and [62]....
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...Several recent papers work in this direction, using especially uninorms [14], [59], [62] but also other aggregation functions [57], [58]....
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...from uninorms have been already studied (see [14] and...
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Book Chapter•DOI•

Fuzzy Set-Theoretic Operators and Quantifiers

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János Fodor, Ronald R. Yager

01 Jan 2000

TL;DR: This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets and other operations which have no counterpart in the classical theory but play some important role in fuzzy sets.

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Abstract: This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets Since these extensions are mainly pointwisely defined, we review basic results on the underlying unary or binary operations on the unit interval such as negations, t-norms, t-conorms, implications, coimplications and equivalences Some strongly related connectives (means, OWA, weighted, and prioritized operations) are also considered, emphasizing the essential differences between these and the formerly investigated operator classes We also show other operations which have no counterpart in the classical theory but play some important role in fuzzy sets (like symmetric sums, weak t-norms and conorms, compensatory AND)

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114 citations

Journal Article•

Residual implications and co-implications from idempotent uninorms

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Daniel Ruiz, Joan Torrens

01 Jan 2004-Kybernetika

TL;DR: The study of implication (and co-implication) functions defined from idempotent uninorms, a list of their properties, as well as some particular cases are studied.

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Abstract: This paper is devoted to the study of implication (and co-implication) functions defined from idempotent uninorms. The expression of these implications, a list of their properties, as well as some particular cases are studied. It is also characterized when these implications satisfy some additional properties specially interesting in the framework of implication functions, like contrapositive symmetry and the exchange principle.

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101 citations

Journal Article•DOI•

A map of dependencies among three-valued logics

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Davide Ciucci¹, Davide Ciucci², Didier Dubois²•Institutions (2)

University of Milan¹, Paul Sabatier University²

20 Nov 2013-Information Sciences

TL;DR: It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic, which can be instrumental when choosing the appropriate fragment for each application context, based on the appropriate meaning of the third truth-value.

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99 citations

Cites background from "Residual operators of uninorms"

...Here, the interpretation given to the third value is irrelevant: moreover, the (conjunctive) discrete uninorm ⁄2 has implication ?2 for its residuum [5,43] (as also discussed later)....
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Journal Article•DOI•

Defending against strategic manipulation in uninorm-based multi-agent decision making

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Ronald R. Yager¹•Institutions (1)

Iona College¹

16 Aug 2002-European Journal of Operational Research

TL;DR: The possible use of the uninorm aggregation operator is described as a way of combining individual agents' preference functions to obtain a group preference function and a mechanism is suggested for modifying the construction of the group decision function to defend against this type of strategic manipulation.

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95 citations

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References

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Book•

Probabilistic Metric Spaces

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Berthold Schweizer, A. Sklar

01 Jan 1983

2,631 citations

Book•

Fuzzy Preference Modelling and Multicriteria Decision Support

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János Fodor, Marc Roubens

31 Oct 1994

TL;DR: This dissertation aims to provide a history of web exceptionalism from 1989 to 2002, a period chosen in order to explore its roots as well as specific cases up to and including the year in which descriptions of “Web 2.0” began to circulate.

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Abstract: Introduction. 1. Fuzzy logical connectives. 2. Valued binary relations. 3. Valued preference modelling. 4. Similarity relations and valued orders. 5. Aggregation operations. 6. Ranking procedures. 7. Multiple criteria decision making. 8. Summary, perspectives and open problems. Index.

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1,886 citations

"Residual operators of uninorms" refers background in this paper

...These classes of operations are mathematically sound and contain a wide variety of particular members (see [14, 18, 29])....
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Journal Article•DOI•

A review of fuzzy set aggregation connectives

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Didier Dubois¹, Henri Prade¹•Institutions (1)

Paul Sabatier University¹

01 Jul 1985-Information Sciences

TL;DR: An extensive survey on fuzzy set-theoretic operations is provided, and the relevance of the theory of functional equations in the axiomatical construction of classes of such operations and the derivation of functional representations is emphasized.

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932 citations

"Residual operators of uninorms" refers background in this paper

...These classes of operations are mathematically sound and contain a wide variety of particular members (see [14, 18, 29])....
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Journal Article•DOI•

Uninorm aggregation operators

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Ronald R. Yager¹, Alexander Rybalov²•Institutions (2)

Iona College¹, The Graduate Center, CUNY²

27 May 1996-Fuzzy Sets and Systems

TL;DR: A generalization of the t-norm and t-conorm called the uni-norm is defined and a class of operators called RQ-star aggregation operators which are useful for aggregations guided by imperatives such as “if most of the scores are above the identity take the Max else use the Min” are introduced.

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861 citations

"Residual operators of uninorms" refers background or methods in this paper

...Such situations can perfectly be modelled by uninorms and this leads to the particular classes introduced in [30], and also to the general forms studied in [19]....
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...Two semantical ways of expressing aggregation of the obtained numbers are as follows [30]:...
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...Definition 1 [30] A uninorm U is an increasing, associative and commutative binary operator that satisfies...
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...In fact, these are the first uninorms considered by Yager and Rybalov [30]....
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...90 3 Uninorms and their structure Uninorms have been introduced recently by Yager and Rybalov as a generalization of both t-norms and t-conorms....
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Journal Article•DOI•

Structure of uninorms

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János Fodor¹, Ronald R. Yager², Alexander Rybalov³•Institutions (3)

University of Agricultural Sciences, Dharwad¹, Iona College², The Graduate Center, CUNY³

01 Aug 1997-International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

TL;DR: An exhaustive study of uninorm operators is established and it is shown that uninorms can be built up from t-norms and t-conorms by a construction similar to ordinal sums.

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Abstract: An exhaustive study of uninorm operators is established. These operators are generalizations of t-norms and t-conorms allowing the neutral element lying anywhere in the unit interval. It is shown that uninorms can be built up from t-norms and t-conorms by a construction similar to ordinal sums. De Morgan classes of uninorms are also described. Representability of uninorms is characterized and a general representation theorem is proved. Finally, pseudo-continuous uninorms are defined and completely classified.

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622 citations

"Residual operators of uninorms" refers background in this paper

...Proposition 1 [19] Consider a uninorm U with neutral element e....
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...Such situations can perfectly be modelled by uninorms and this leads to the particular classes introduced in [30], and also to the general forms studied in [19]....
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...[19] have investigated the existence of uninorms with a similar representation in terms of a single-variable function....
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...Proposition 2 [19] Consider a uninorm U with neutral element e, then...
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...Proof: The implication from left to right was shown in [19]....
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