# On the fundamentals of fuzzy sets.

Stochastica (1984)

- Volume: 8, Issue: 2, page 157-169
- ISSN: 0210-7821

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topLowen, Robert. "On the fundamentals of fuzzy sets.." Stochastica 8.2 (1984): 157-169. <http://eudml.org/doc/38899>.

@article{Lowen1984,

abstract = {A considerable amount of research has been done on the notions of pseudo complement, intersection and union of fuzzy sets [1], [4], [11]. Most of this work consists of generalizations or alternatives of the basic concepts introduced by L. A. Zadeh in his famous paper [13]: generalization of the unit interval to arbitrary complete and completely distributive lattices or to Boolean algebras [2]; alternatives to union and intersection using the concept of t-norms [3], [10]; alternative complements in the unit interval in [4], [11], a.s.o. Although it is usually accepted that I := [0,1] is a natural generalization of \{0,1\} and that a --> 1 - a is a natural generalization of the Boolean complement on \{0,1\}, we do however not find canonical and mathematical justification for this fact, which nevertheless lies at the heart of the definition of fuzzy sets. It is the purpose of this note to present a canonical way of obtaining I and L. A. Zadeh's pseudocomplement.Moreover, if we consistently use this canonical machine we shall see that also other set-concepts can, and maybe should have been extended to fuzzy sets. Further we also give a possible generalization involving a choice of arbitrary t-norms.},

author = {Lowen, Robert},

journal = {Stochastica},

keywords = {Conjuntos difusos; Estudio teórico; pseudo complement; intersection; union; fuzzy sets; t-norms},

language = {eng},

number = {2},

pages = {157-169},

title = {On the fundamentals of fuzzy sets.},

url = {http://eudml.org/doc/38899},

volume = {8},

year = {1984},

}

TY - JOUR

AU - Lowen, Robert

TI - On the fundamentals of fuzzy sets.

JO - Stochastica

PY - 1984

VL - 8

IS - 2

SP - 157

EP - 169

AB - A considerable amount of research has been done on the notions of pseudo complement, intersection and union of fuzzy sets [1], [4], [11]. Most of this work consists of generalizations or alternatives of the basic concepts introduced by L. A. Zadeh in his famous paper [13]: generalization of the unit interval to arbitrary complete and completely distributive lattices or to Boolean algebras [2]; alternatives to union and intersection using the concept of t-norms [3], [10]; alternative complements in the unit interval in [4], [11], a.s.o. Although it is usually accepted that I := [0,1] is a natural generalization of {0,1} and that a --> 1 - a is a natural generalization of the Boolean complement on {0,1}, we do however not find canonical and mathematical justification for this fact, which nevertheless lies at the heart of the definition of fuzzy sets. It is the purpose of this note to present a canonical way of obtaining I and L. A. Zadeh's pseudocomplement.Moreover, if we consistently use this canonical machine we shall see that also other set-concepts can, and maybe should have been extended to fuzzy sets. Further we also give a possible generalization involving a choice of arbitrary t-norms.

LA - eng

KW - Conjuntos difusos; Estudio teórico; pseudo complement; intersection; union; fuzzy sets; t-norms

UR - http://eudml.org/doc/38899

ER -

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