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An extension of several properties for fuzzy t-norm and vague t-norm
Journal of Intelligent & Fuzzy SystemsRosenfeld defined a fuzzy subgroup of a given group as a fuzzy subset with two special conditions and Mustafa Demirci proposed the idea of fuzzifying the operations on a group through a fuzzy equality and a fuzzy equivalence relation. This paper mainly focuses on fuzzy subsets and vague sets of monoids with several extended algebraic properties ...
Wang, Haohao, Li, Wei, Yang, Bin
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Regular left-continuous t-norms
Semigroup Forum, 2008A triangular norm (t-norm) \(\odot\) can be determined by its \textit{translations} \(\lambda_a:[0,1]\to[0,1]\), \(x\mapsto x\odot a\), where \(a\in[0,1]\). They form the set \(\Lambda=\{\lambda_a: a\in[0,1]\}\) of (non-strictly) increasing, mutually commuting functions and for each \(a\in[0,1]\), \(\lambda_a\) is the unique element of \(\Lambda\) such
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Vector t-Norms With Applications
IEEE Transactions on Fuzzy Systems, 2017Some basic t-norms defined on [0, 1] are well known in many study areas and applications. However, more general extension of them into vector forms can be used in a lot of new decision-making realms. In this study, we first define preference vector on a linearly ordered set, which includes different special vectors that are mathematically equivalent ...
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2011 International Conference on Electrical and Control Engineering, 2011
In fuzzy logic in wider sense, i.e. in the field of fuzzy sets applications, t-norms got a prominent role in recent times. In many-valued logic, the ŁUKASIEWICZ systems, and also the product logic all are t-norm based systems. The main emphasis in the present paper is on propositional logic.
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In fuzzy logic in wider sense, i.e. in the field of fuzzy sets applications, t-norms got a prominent role in recent times. In many-valued logic, the ŁUKASIEWICZ systems, and also the product logic all are t-norm based systems. The main emphasis in the present paper is on propositional logic.
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On the determination of left-continuous t-norms and continuous archimedean t-norms on some segments
Aequationes mathematicae, 2005A \(t\)(riangular) norm is a function \(T:[0,1]^2\to [0,1]\) such that for all \(x,y,z\in [0,1]\) we have \(T(x,y)=T(y,x),\,T(T(x,y),z)=T(x,T(y,z)),\,T(x,1)=x\) and \(T(x,\cdot)\) is increasing. The author finds new sets of uniqueness for (continuous Archimedean and left continuous) \(t\)-norms. These sets of uniqueness are some vertical segments of \([
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Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices
Fuzzy Sets and Systems, 2012Ordinal sums of triangular norms (t-norms) on bounded lattices are studied. Originally, t-norms were defined on the unit interval. Later, a generalization on a more general algebraic structure, namely bounded lattices, was proposed. It turned out that in this case, an ordinal sum in Clifford's sense of t-norms may not be a t-norm.
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Probabilistic Prototype Classification Using t-norms
2014We introduce a generalization of Multivariate Robust Soft Learning Vector Quantization. The approach is a probabilistic classifier and can deal with vectorial class labelings for the training data and the prototypes. It employs t-norms, known from fuzzy learning and fuzzy set theory, in the class label assignments, leading to a more flexible model with
Tina Geweniger +2 more
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1978
A t-norm is a mapping T: [0, 1] × [0, 1] → [0, 1] such that: (i) T(a, 1) = a; (ii) T(a, b) = T(b, a); (iii) T(a, b) ≤ T(c, d), whenever a ≤ c, b≤ d; (iv) T(a, T(b, c)) = T(T(a, b), c).
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A t-norm is a mapping T: [0, 1] × [0, 1] → [0, 1] such that: (i) T(a, 1) = a; (ii) T(a, b) = T(b, a); (iii) T(a, b) ≤ T(c, d), whenever a ≤ c, b≤ d; (iv) T(a, T(b, c)) = T(T(a, b), c).
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2011
Recently, in [3], we have discussed the distributive equation of implications I(x, T1(y, z)) = T2(I(x, y), I(x, z)) over t-representable t-norms generated from strict t-norms in interval-valued fuzzy sets theory. In this work we continue these investigations, but for t-representable t-norms generated from nilpotent t-norms.
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Recently, in [3], we have discussed the distributive equation of implications I(x, T1(y, z)) = T2(I(x, y), I(x, z)) over t-representable t-norms generated from strict t-norms in interval-valued fuzzy sets theory. In this work we continue these investigations, but for t-representable t-norms generated from nilpotent t-norms.
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