Topological structures in fuzzy lattices
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 equivalent to Hutton's definition.
syntopogeneous structures on fuzzy lattices
scientific article; zbMATH DE number 3967391
scientific article