Topology on residuated lattices for biresiduum-based continuity of fuzzy sets

By continuity of [0,1]-valued fuzzy set it is usually understood standard continuity of a real-valued function. This could be in conflict with the similarity fuzzy relation on [0,1] given by the operation of biresiduum induced by a left-continuous t-norm. In this work, a new kind of continuity, call...

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Published inFuzzy sets and systems Vol. 498; p. 109133
Main Author Krupka, Michal
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.01.2025
Subjects
Continuity
Fuzzy set
Order dimension
Residuated lattice
Scott-open set
Topology
Continuity
Topology
Order dimension
Scott-open set
Fuzzy set
Residuated lattice
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Summary:By continuity of [0,1]-valued fuzzy set it is usually understood standard continuity of a real-valued function. This could be in conflict with the similarity fuzzy relation on [0,1] given by the operation of biresiduum induced by a left-continuous t-norm. In this work, a new kind of continuity, called t-continuity, of an L-set in a topological space for L being a complete residuated lattice of finite dimension is defined. It is based on a notion of oscillation of L-set, which is an element of L computed by means of biresiduum in L. A topology on L of this continuity, called the t-topology on L, is found. This topology is constructed by means of biresiduum and a variant of Scott-open sets called residuated Scott-open sets. It is shown that the t-topology of L is Hausdorff and that L together with its t-topology is a topological residuated lattice. T-topologies of four standard t-norms are found. The notion of t-topology is generalized to sets with L-equivalence relation and a new characteristic of extensional sets implying that extensional L-sets are t-continuous is given.
ISSN:0165-0114
DOI:10.1016/j.fss.2024.109133