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 f-image 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 f-set. 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.
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