Fuzzy Galois connections on fuzzy posets

Yao, Wei; Lü, Lei

doi:10.1002/malq.200710079

Cited by 94 publications

(42 citation statements)

References 18 publications

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“…Basics of the theory of isotone Galois connections come from [21], other references are [22,16,15]. An isotone L-Galois connection between L-ordered sets U and V is a pair f, g where f :…”

Section: Isotone L-galois Connectionsmentioning

confidence: 99%

Complete relations on fuzzy complete lattices

Konečný

Krupka

2017

Fuzzy Sets and Systems

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We generalize the notion of complete binary relation on complete lattice to residuated lattice valued ordered sets and show its properties. Then we focus on complete fuzzy tolerances on fuzzy complete lattices and prove they are in one-to-one correspondence with extensive isotone Galois connections. Finally, we prove that any fuzzy complete lattice factorized by a complete fuzzy tolerance is again a fuzzy complete lattice.

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“…Basics of the theory of isotone Galois connections come from [21], other references are [22,16,15]. An isotone L-Galois connection between L-ordered sets U and V is a pair f, g where f :…”

Section: Isotone L-galois Connectionsmentioning

confidence: 99%

Complete relations on fuzzy complete lattices

Konečný

Krupka

2017

Fuzzy Sets and Systems

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“…We say that (f, g) is a Q-adjunction, or a (Q-monotone) Galois connection [10], if e Y (f (x), y) = e X (x, g(y)). Then, f is called a left and g a right Q-adjoint.…”

Section: Definition 34mentioning

confidence: 99%

“…We omit the proof as it proceeds exactly in the same way as the one in [10], only considering the base quantale Q rather than a complete residuated lattice L (in a residuated lattice 1 = ). Proposition 3.13.…”

Section: Q-sup-latticesmentioning

confidence: 99%

“…A number of papers on the topic of sets with fuzzy order relations valuated in a complete lattice with additional structure have been published in recent years. Among many others, articles [9,10,12] may be used as a reference. There are various structures used for fuzzy valuation in the literature, e.g., frames [3] and complete residuated lattices [12], both being just special cases of quantales.…”

Section: Introductionmentioning

confidence: 99%

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On some basic constructions in categories of quantale-valued sup-lattices

Šlesinger¹

2016

Math. Appl.

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Abstract. If the standard concepts of partial-order relation and subset are fuzzified, taking valuation in a unital commutative quantale Q, corresponding concepts of joins and join-preserving mappings can be introduced. We present constructions of limits, colimits and Hom-objects in categories Q-Sup of Q-valued fuzzy joinsemilattices, showing the analogy to the ordinary category Sup of join-semilattices. IntroductionIn the standard concept of a fuzzy set [11], the relation "x is an element of X" is fuzzified, and replaced by a mapping X → [0, 1] assigning to each element of X its "membership degree". As further generalization, the membership degree can be evaluated in structures more general than the real unit interval, typically frames, residuated lattices, or quantales.In our paper, the concept of a set remains unchanged, and it will be the partial order relation and the notion of a subset that will be replaced by suitable mappings to a quantale. Instead of considering on X a partial order relation ≤, we employ a unital quantale Q and mappings M : X → Q and e : X × X → Q, which quantify the "degree of truth" of membership in a subset and of being less or equal.Sets equipped with quantale-valued binary mappings were initially investigated in the so-called quantitative domain theory [4]. A number of papers on the topic of sets with fuzzy order relations valuated in a complete lattice with additional structure have been published in recent years. Among many others, articles [9,10,12] may be used as a reference. There are various structures used for fuzzy valuation in the literature, e.g., frames [3] and complete residuated lattices [12], both being just special cases of quantales. Also terminology has not settled yet, and differs among authors. As the multiplicative unit of a quantale need not be its top element, even truth can have more degrees in Q. This is different from valuation using frames or residuated lattices where the unit is the top element as well.The basic properties of the category of complete join-semilattices as well as the fundamental constructions in this category such as limits and colimits have MSC (2010): primary 08A72, 06F99; secondary 18B35.

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“…A Galois connection is an important mathematical tool for algebraic structure, data analysis, and knowledge processing [1][2][3][4][5][6][7][8][9][10]. Hájek [11] introduced a complete residuated lattice L that is an algebraic structure for many-valued logic.…”

Section: Introductionmentioning

confidence: 99%