Complete relations on fuzzy complete lattices
Jan Konecny, Michal Krupka
TL;DR
This paper extends the concept of complete binary relations to fuzzy complete lattices, establishing their properties and connections with Galois connections, and demonstrating the preservation of completeness under fuzzy tolerances.
Contribution
It introduces a generalization of complete relations to fuzzy lattices and links fuzzy tolerances with Galois connections, showing the preservation of lattice completeness.
Findings
Complete fuzzy tolerances correspond to extensive isotone Galois connections.
Fuzzy complete lattices remain complete after factorization by a fuzzy tolerance.
The paper generalizes classical lattice relations to the fuzzy setting.
Abstract
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 fuzzy complete lattice, factorized by a complete fuzzy tolerance, is again a fuzzy complete lattice.
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