Residual implications and left-continuous t-norms which are ordinal sums of semigroups

Mesiar, Radko; Mesiarová, Andrea

doi:10.1016/j.fss.2003.06.008

Cited by 44 publications

(14 citation statements)

References 12 publications

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“…and Mesiarová (2004) we have (a → T b) → T b = a if and only if e ( a e .Because of one-to-one-correspondence we have the required result.…”

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confidence: 89%

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Semi-divisible triangular norms

Mesiarová-Zemánková

2007

Soft Comput

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Semi-divisibility of left-continuous triangular norms is a weakening of the divisibility (i.e., continuity) axiom for t-norms. In this contribution we focus on the class of semi-divisible t-norms and show the following properties: Each semi-divisible t-norm with Ran(n T ) = [0, 1] is nilpotent. Semi-divisibility of an ordinal sum t-norm is determined by the corresponding property of its first component (which can be a proper t-subnorm, too). Finally, negations with finite range derived from semi-divisible t-norms are studied.

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“…and Mesiarová (2004) we have (a → T b) → T b = a if and only if e ( a e .Because of one-to-one-correspondence we have the required result.…”

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confidence: 89%

“…However, the situation is different in the case when the first summand is a t-subnorm. This follows from the fact that in general the residual implications of t-subnorms do not fulfil the equality 1 → S a = a and the property a → S b = 1 if and only if a ≤ b (see Mesiar and Mesiarová 2004).…”

Section: Examplementioning

confidence: 99%

Semi-divisible triangular norms

Mesiarová-Zemánková

2007

Soft Comput

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“…The following method is a kind of generalization of the results obtained, e.g., in [7] for residual implications. Figure 4.…”

Section: Ordinal Sums Of Fuzzy Implicationsmentioning

confidence: 99%

“…Some interesting results connected to representation of the residual implication corresponding to a fuzzy conjunction (for example continuous or at least left-continuous tnorm) given by an ordinal sum were obtained in [2], [4], [7]. In [9] S u et al introduced a concept of an ordinal sum of fuzzy implications similar to the construction of the ordinal sum of t-norms.…”

Section: Introductionmentioning

confidence: 99%

Generating Fuzzy Implications by Ordinal Sums

Drygaś

Król

2016

Tatra Mountains Mathematical Publications

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ABSTRACT. This paper deals with ordinal sums of fuzzy implications. Some of the known constructions are recalled and new ways of generating fuzzy implications from given ones are proposed. Sufficient properties of fuzzy implications as summands for obtaining a fuzzy implication as a result are presented. IntroductionFuzzy implications find applications in many fields such as fuzzy control, approximate reasoning, and decision support systems. For that reason, new families of these connectives are the subject of investigation.One of the directions of such a research is to consider an ordinal sum of fuzzy implications on the pattern of the ordinal sum of t-norms. Some interesting results connected to representation of the residual implication corresponding to a fuzzy conjunction (for example continuous or at least left-continuous tnorm) given by an ordinal sum were obtained in [2], [4], [7]. In [9] S u et al. introduced a concept of an ordinal sum of fuzzy implications similar to the construction of the ordinal sum of t-norms. In [3], there were proposed other constructions of ordinal sums of fuzzy implications. In this contribution, some of the ideas are recalled and new possibilities of defining ordinal sums of fuzzy implications are proposed.First, in Section 2, we recall definitions and basic results concerning t-norms and fuzzy implications and we revise methods of constructing ordinal sums of fuzzy implications. Then, in Section 3, we propose new constructions of ordinal sums of fuzzy implication which generate fuzzy implications. Next, we suggest further research directions for the ordinal sums of fuzzy implications.

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“…By [10], a continuous t-norm has an ordinal sum structure (]a k , b k [, k , k ∈ K), where the summands are continuous Archimedean t-norms (that is x x = x iff x ∈ {0, 1}; obviously Lukasiewicz t-norm and Product t-norms are Archimedean while Gödel t-norm is not), for detail of this representation, see Theorem 1 in [11]. Moreover, by [2], the residuum of a continuous t-norm with Archimedean summands has related ordinal sum structure, too.…”

Section: Definition 1 (Klement Et Al [10]) An Additive Generator F mentioning

confidence: 99%