# Fuzzy relation equation under a class of triangular norms: A survey and new results.

Antonio Di Nola; Witold Pedrycz; Salvatore Sessa; Wang Pei Zhuang

Stochastica (1984)

- Volume: 8, Issue: 2, page 99-145
- ISSN: 0210-7821

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topDi Nola, Antonio, et al. "Fuzzy relation equation under a class of triangular norms: A survey and new results.." Stochastica 8.2 (1984): 99-145. <http://eudml.org/doc/38901>.

@article{DiNola1984,

abstract = {By substituting the classical lattice operator min of the unit real interval with a triangular norm of Schweizer and Sklar, the usual fuzzy relational equations theory of Sanchez can be generalized to wider theory of fuzzy equations. Considering a remarkable class of triangular norms, for such type of equations defined on finite sets, we characterize the upper an lower solutions.We also characterize the solutions posessing a minimal fuzziness measure of Yager valued with respect to a triangular norm and conorm.Moreover we discuss the problem of characterization of the approximate solutions of fuzzy equations.Finally, the role of the equations considered here in creation of a formal framework for copying with fuzziness is illustrated by various examples in some well known schemes in applications of fuzzy set theory.},

author = {Di Nola, Antonio, Pedrycz, Witold, Sessa, Salvatore, Pei Zhuang, Wang},

journal = {Stochastica},

keywords = {Conjuntos difusos; Probabilidad; Ecuaciones; Norma triangular; fuzzy number; possibility measure; generalized fuzzy relation equations; minimal fuzziness measure; approximate solutions; fuzzy equations; copying with fuzziness},

language = {eng},

number = {2},

pages = {99-145},

title = {Fuzzy relation equation under a class of triangular norms: A survey and new results.},

url = {http://eudml.org/doc/38901},

volume = {8},

year = {1984},

}

TY - JOUR

AU - Di Nola, Antonio

AU - Pedrycz, Witold

AU - Sessa, Salvatore

AU - Pei Zhuang, Wang

TI - Fuzzy relation equation under a class of triangular norms: A survey and new results.

JO - Stochastica

PY - 1984

VL - 8

IS - 2

SP - 99

EP - 145

AB - By substituting the classical lattice operator min of the unit real interval with a triangular norm of Schweizer and Sklar, the usual fuzzy relational equations theory of Sanchez can be generalized to wider theory of fuzzy equations. Considering a remarkable class of triangular norms, for such type of equations defined on finite sets, we characterize the upper an lower solutions.We also characterize the solutions posessing a minimal fuzziness measure of Yager valued with respect to a triangular norm and conorm.Moreover we discuss the problem of characterization of the approximate solutions of fuzzy equations.Finally, the role of the equations considered here in creation of a formal framework for copying with fuzziness is illustrated by various examples in some well known schemes in applications of fuzzy set theory.

LA - eng

KW - Conjuntos difusos; Probabilidad; Ecuaciones; Norma triangular; fuzzy number; possibility measure; generalized fuzzy relation equations; minimal fuzziness measure; approximate solutions; fuzzy equations; copying with fuzziness

UR - http://eudml.org/doc/38901

ER -

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