On the structure of left-continuous t-norms that have a continuous contour line

Maes, Koen; Baets, Bernard De

doi:10.1016/j.fss.2006.09.016

Cited by 43 publications

(12 citation statements)

References 21 publications

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“…(i) It is easy to prove, that both N T and N S are fuzzy negations. In the literature N T is also called the contour line C 0 of T, while N S is called the contour line D 1 of S (see [24,26]). …”

Section: Remark 25mentioning

confidence: 99%

“…For the methods of obtaining these families see the works of Jenei [16,17,19,20] connected with the rotation and the rotation-annihilation and the works of Maes and De Baets [25,26,28] connected with the triple rotation. In fact, it can be shown that every t-norm (T MD ) can be obtained as a rotation-annihilation of a particular ordinal sum of the Łukasiewicz t-norm T LK (see [25, p. 384]).…”

mentioning

confidence: 99%

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- and R-implications: A state-of-the-art survey

Baczyński

Jayaram

2008

Fuzzy Sets and Systems

139

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Section: Remark 25mentioning

confidence: 99%

mentioning

confidence: 99%

- and R-implications: A state-of-the-art survey

Baczyński

Jayaram

2008

Fuzzy Sets and Systems

139

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“…2 , with N T the residual negator of T (see [23] for more information on the residual negator). For the relationship of this notion with S-completeness, we refer to [3].…”

Section: Complete Fuzzy Pre-ordersmentioning

confidence: 99%

On the compositional characterization of complete fuzzy pre-orders

Díaz

Baets

Montes

2008

Fuzzy Sets and Systems

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Complete pre-orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations. In this paper, we investigate this characterization in a fuzzy setting. We consider two types of completeness (weak completeness and strong completeness) and decompose a fuzzy pre-order by means of an indifference generator, in particular a Frank t-norm. In the weakly complete case, we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them. In the strongly complete case, we lay bare a stronger type of transitivity of the strict preference relation. We conclude the paper with a rather negative result: there is no hope to obtain a compositional characterization of weakly complete fuzzy pre-orders, and hence also not of fuzzy pre-orders in general.keyword Frank t-norm, fuzzy pre-order, indifference, strict preference, transitivity

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“…For another characterization of pseudo t-norms, see also [27] and [28]. In particular, we derive the following characterization of left-continuous t-norms.…”

Section: Proposition 33 ([5]mentioning

confidence: 99%

Conjunctors and their Residual Implicators: Characterizations and Construction Methods

Durante

Klement

Mesiar

et al. 2007

MedJM

121

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In many practical applications of fuzzy logic it seems clear that one needs more flexibility in the choice of the conjunction: in particular, the associativity and the commutativity of a conjunction may be removed. Motivated by these considerations, we present several classes of conjunctors, i.e. binary operations on [0, 1] that are used to extend the boolean conjunction from {0, 1} to [0, 1], and characterize their respective residual implicators. We establish hence a one-to-one correspondence between construction methods for conjunctors and construction methods for residual implicators. Moreover, we introduce some construction methods directly in the class of residual implicators, and, by using a deresiduation procedure, we obtain new conjunctors.

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On the structure of left-continuous t-norms that have a continuous contour line

Cited by 43 publications

References 21 publications

- and R-implications: A state-of-the-art survey

- and R-implications: A state-of-the-art survey

On the compositional characterization of complete fuzzy pre-orders

Conjunctors and their Residual Implicators: Characterizations and Construction Methods

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