Abstract
This paper is the third in a series of papers studying equivalence classes of fuzzy subgroups of a given group under a suitable equivalence relation. We introduce the notion of a pinned flag in order to study the operations sum, intersection and union, and their behavior with respect to the equivalence. Further, we investigate the extent to which a homomorphism preserves the equivalence. Whenever the equivalences are not preserved, we have provided suitable counterexamples.
Highlights
For the benefit of the reader and to fix notations, we recall the following from [3].We use I = [0, 1], the real unit interval, as a chain with the usual ordering in which ∧ stands for infimum and ∨ stands for supremum.A fuzzy subset of a set X is a mapping μ : X → I
If X is a finite group, a fuzzy set μ is said to be a fuzzy subgroup if μ(x + y) ≥ μ(x) ∧ μ(y) for all x, y ∈ X and μ(x) = μ(−x)
Given a homomorphism between two groups, we look at the equivalence classes of homomorphic images and preimages of fuzzy subgroups
Summary
We associate the following fuzzy subgroup with such a pinned flag (Ꮿ, l): Given a homomorphism between two groups, we look at the equivalence classes of homomorphic images and preimages of fuzzy subgroups. We may have inequivalent fuzzy subgroups giving rise to equivalent preimages under a homomorphism.
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