The Dedekind–MacNeille completions for fuzzy posets

Wei-xian, Xie; Zhang, Qiye; Fan, Lei

doi:10.1016/j.fss.2008.12.002

Cited by 32 publications

(5 citation statements)

References 10 publications

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“…Remark 2.6 Notice that Proposition 2.5 provides different ways to define the join and meet of L — sets. As a matter of fact, this has been shown in 21 where L is a frame as the truth value structure (note: every frame can be viewed as a complete residuated lattice), and the notion of L —fuzzy complete lattice introduced there is consistent with Bělohlávek’s completely lattice L —ordered set (see 6).…”

Section: L—ordered Sets and Fuzzy Closure Operatorsmentioning

confidence: 85%

“…Proposition 2.5 6, 21, 27 Let (( X , ≈), e ) be an L —ordered set and A ∈ L X . The following are equivalent: A inf (respectively, A sup ) is an L —singleton. There exists (unique) x 0 ∈ X such that A inf ( x 0 )=1 (respectively, A sup ( x 0 )=1). There exists (unique) x 0 ∈ X such that A ( x ) ≤ e ( x 0 , x ) (respectively, A ( x ) ≤ e ( x , x 0 )) for any x ∈ X , and \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\bigwedge _{x\in X}(A(x)\rightarrow e(y, x))\le e(y, x_{0}) $\end{document} (respectively, \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\bigwedge _{x\in X}(A(x)\rightarrow e(x, y))\le e(x_{0}, y) $\end{document}) for any y ∈ X . …”

Section: L—ordered Sets and Fuzzy Closure Operatorsmentioning

confidence: 99%

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Fuzzy closure systems onL-ordered sets

Guo

Zhang

2011

Mathematical Logic Quarterly

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MSC (2010) 03E72, 06A15In this paper, notions of fuzzy closure system and fuzzy closure L-system on L-ordered sets are introduced from the fuzzy point of view. We first explore the fundamental properties of fuzzy closure systems. Then the correspondence between fuzzy closure systems (fuzzy closure L-systems) and fuzzy closure operators is established. Finally, we study the connections between fuzzy closure systems and fuzzy Galois connections.

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Section: L—ordered Sets and Fuzzy Closure Operatorsmentioning

confidence: 85%

Section: L—ordered Sets and Fuzzy Closure Operatorsmentioning

confidence: 99%

Fuzzy closure systems onL-ordered sets

Guo

Zhang

2011

Mathematical Logic Quarterly

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“…The fruitful results about completions of crisp order structures have inspired some scholars to study the completions in the fuzzy setting. Wagner [14], Bȇlohlávek [15,16] and Xie [17] generalized the DM-completion to the multi-valued framework from different perspectives. The interesting thing is that the DM-completions above are exactly the same when Ω = L is a frame.…”

Section: Introductionmentioning

confidence: 99%

ZL-Completions for ZL-Semigroups

2022

Symmetry

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In this paper, we generalize a common completion pattern of ordered semigroups to the fuzzy setting. Based on a standard L-completion ZL, we introduce the notion of a ZL-semigroup as a generalization of an L-ordered semigroup, where L is a complete residuated lattice. For this asymmetric mathematical structure, we define a ZL-completion of it to be a complete residuated L-ordered semigroup together with a join-dense L-ordered semigroup embedding satisfying the universal property. We prove that: (1) For every compositive ZL, the category CSL of complete residuated L-ordered semigroups is a reflective subcategory of the category SZL of ZL-semigroups; (2) for an arbitrary ZL, there is an adjunction between SZL and the category SZL→E of weakly ZL-continuous L-ordered semigroup embeddings of ZL-semigroups. By appropriate specialization of ZL, the results can be applied to the DML-completion, certain completions associated with fuzzy subset systems, etc.

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“…Then, Venugopalan [2] developed a structure of fuzzy ordered sets. Since then, many authors have studied fuzzy relations and ordering by using different approaches [3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

A Study on Fuzzy Order Bounded Linear Operators in Fuzzy Riesz Spaces

et al. 2021

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This paper aims to study fuzzy order bounded linear operators between two fuzzy Riesz spaces. Two lattice operations are defined to make the set of all bounded linear operators as a fuzzy Riesz space when the codomain is fuzzy Dedekind complete. As a special case, separation property in fuzzy order dual is studied. Furthermore, we studied fuzzy norms compatible with fuzzy ordering (fuzzy norm Riesz space) and discussed the relation between the fuzzy order dual and topological dual of a locally convex solid fuzzy Riesz space.

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The Dedekind–MacNeille completions for fuzzy posets

Cited by 32 publications

References 10 publications

Fuzzy closure systems onL-ordered sets

Fuzzy closure systems onL-ordered sets

ZL-Completions for ZL-Semigroups

A Study on Fuzzy Order Bounded Linear Operators in Fuzzy Riesz Spaces

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