On extensions of triangular norms on bounded lattices

Saminger‐Platz, Susanne; Klement, Erich Peter; Mesiar, Radko

doi:10.1016/s0019-3577(08)80019-5

Cited by 69 publications

(26 citation statements)

References 17 publications

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“…One of the first published works on this subject was put forward by Saminger-Platz et al in [1]. There, it was proposed a method to extend a given t-norm T defined on a complete ordinary sublattice M of lattice L.…”

Section: Extension Methods Via Retractions (Emr)mentioning

confidence: 99%

“…is known that, given a lattice L, the relation x ≤ L y if and only if x ∧ L y = x defines a partial order on L. Recall also that a lattice L is called a complete lattice if every subset of it has a supremum and an infimum element 1 .…”

Section: Bounded Lattices: Definition and Related Conceptsmentioning

confidence: 99%

“…The pioneer work in this framework was put forward by Saminger-Platz et al in [1] which provides a method to extend a t-norm T from a complete sublattice M to a bounded lattice L. Later, *Correspondence: espalmeira@uesc.br 1 Departamento de Ciências Exatas e Tecnológicas -DCET, Universidade…”

Section: Introductionmentioning

confidence: 99%

“…Estadual de Santa Cruz -UESC, Campus Soane Nazaré de Andrade, Rodovia Jorge Amado, Km 16, Bairro Salobrinho, 45662-900, Ilhéus, Brazil Full list of author information is available at the end of the article we have developed in [2] an extension method to extend t-norms, t-conorms, and fuzzy negations that generalizes the method proposed in [1] considering a modified notion of sublattice. Also, we have applied this method for fuzzy implications in [3].…”

Section: Introductionmentioning

confidence: 99%

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Some results on extension of lattice-valued QL-implications

2016

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Background: A very important issue in lattice theory is how to extend a given operator preserving its algebraic properties. For lattice-valued fuzzy operators framework, in 2008 Saminger-Platz presented a way to extend t-norms which was generalized by Palmeira et al. (2011) for t-norms, t-conorms, fuzzy negations and implications, considering the scenery provided by the (r, s)-sublattice. Methods: In this paper we investigated how to extend QL-implications and which properties of it are preserved by the extension method via retractions (EMR). Results:As results, we proved that properties (LB), (RB), (CC1), (CC2), (CC3), (CC4), (L-NP), (EP) and (IP) are preserved by EMR. Conclusions: However, the extension method via retractions fails in preserving the important properties (NP), (OP), (IBL), (CP), (P) and (LEM).

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Section: Extension Methods Via Retractions (Emr)mentioning

confidence: 99%

Section: Bounded Lattices: Definition and Related Conceptsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some results on extension of lattice-valued QL-implications

2016

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“…Moreover, monotonicity of the final operation should be provided such that appropriate candidates for such sublattices are intervals [a, b] = {x ∈ L | a L x L b} with a L b (see also [38]). Therefore, we define ordinal sums of t-norms on lattices in the following way.…”

Section: Triangular Norms On Bounded Latticesmentioning

confidence: 99%