Topological structures in fuzzy lattices
type: Item
instance of: scholarly article
title: Topological structures in fuzzy lattices
zbMATH Open document ID: 0599.54006
DOI: 10.1016/0022-247X(84)90114-8
author: Peter Hamburg
author: Liviu C. Florescu
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: The aim of this paper is to initiate a synthetic theory of topological structures on a complete sublattice \({\mathcal L}\) of the lattice of fuzzy sets \(I^ X\). Particularly, for \({\mathcal L}=I^ X\), we obtain the above- mentioned structures and, for \({\mathcal L}=\{0,1\}^ X\), we rediscover the classical theory. In this way, the connection between the classical case and the fuzzy case is not merely an analogy but also an organic connection. The fuzzy points are a convenient tool in fuzzy topology. Wong defined that the fuzzy point h belongs to fuzzy set a iff \(h(x)<a(x)\) for every \(x\in X\). We change this relation with a very useful relation \(\prec\), which, in the case \({\mathcal L}=\{0,1\}^ X\), is the classical relation \(\in\). In a complete sublattice \({\mathcal L}\subseteq I^ X\), the equivalent of fuzzy points is the notion of a base of a lattice. We define bases and subbases of a lattice and give some properties and examples in Section 2. In Section 3 we define the syntopogeneous structures on fuzzy lattices. Topologies, proximities, and uniformities are special cases of these structures. Using the bases language, we characterize the perfect, coperfect, and biperfect orders associated with a topogenous order. In Section 4 we examine the topological structures on fuzzy lattices. We define the closure, the interior, and the neighborhoods and give a characterization of topological structures in terms of neighbourhoods of fuzzy points. Section 5 is devoted to a brief presentation of proximities on fuzzy lattices. The fuzzy uniformities were defined by Hutton, using the language of uniform mapping. We define in Section 6 the uniformities on fuzzy lattices in the entourages language. The entourages and the neighbourhoods satisfy axioms similar to those of the classical case; add to these a condition of lower continuity trivially verified in the case \({\mathcal L}=\{0,1\}^ X\). In the case \({\mathcal L}=I^ X\) we prove that our definition is equivale…
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 54A15
Mathematics Subject Classification ID: 06B30
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 3967391
zbMATH Keywords: lattice of fuzzy sets
zbMATH Keywords: fuzzy point
zbMATH Keywords: base of a lattice
zbMATH Keywords: syntopogeneous structures on fuzzy lattices
zbMATH Keywords: Topologies
zbMATH Keywords: proximities
zbMATH Keywords: uniformities
zbMATH Keywords: topogenous order
zbMATH Keywords: closure
zbMATH Keywords: interior
zbMATH Keywords: neighborhoods
zbMATH Keywords: fuzzy uniformities
zbMATH Keywords: entourages
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3967391