A Theory of Lattice-Valued Fuzzy Sets and Fuzzy Maps Between Different Lattice- Valued Fuzzy Sets – Revisited
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Abstract
F-Set Theory is a natural generalization of Goguen's L-Fuzzy Set Theory which itself is a generalization of Zadeh's, both Fuzzy and
Interval Valued Fuzzy Set Theories. It naturally and neatly extends several of the crisp (Sub)Set-Map-Properties to: L-valued f-(sub) sets, f-maps
between L-valued f-sets and M-valued f-sets, where the complete lattice L-may possibly different from the complete lattice M, M-valued fimage
of an L-valued f-subset of the domain L-valued f-set and L-valued f-inverse image of an M-valued f-subset of the co-domain M-valued fset.
However, for several of the results in this theory, the complete homomorphisms are assumed to be one or a combination of: 0-preserving, 0-
reflecting, 1-preserving and 1-reflecting. Further, some of the results use the infinite meet distributivity of the underlying complete lattice of the
domain and/or range f-set.
Now the aim of this paper is: 1. to separate this (these) hypothesis (hypotheses) of preserving/reflecting from the results in F-Set Theory and
restate and prove the corresponding results and 2. to remove the hypothesis of infinite meet distributivity of the underlying complete lattice for
truth values via altogether new proofs and 3. to add several new results that are needed/developed in this process.
Keywords: L-Fuzzy Set, L-Fuzzy Image, L-Fuzzy Inverse Image, Complete Lattice.
Subjclass: Primary 94D05; Secondary 04A72, 03E72, 03B50, 20N25, 54A40
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