Abstract
We deal with topics regarding(λ,μ)-fuzzy subgroups, mainly(λ,μ)-fuzzy cosets and(λ,μ)-fuzzy normal subgroups. We give basic properties of(λ,μ)-fuzzy subgroups and present some results related to(λ,μ)-fuzzy cosets and(λ,μ)-fuzzy normal subgroups.
Highlights
Since Zadeh [1] introduced the concept of a fuzzy set in 1965, various algebraic structures have been fuzzified
In 1971, Rosenfeld [2] introduced the notion of a fuzzy subgroup and initiated the study of fuzzy groups
We present a further investigation into properties of (λ, μ)-fuzzy subgroups, (λ, μ)-fuzzy normal subgroups, and (λ, μ)-fuzzy left and right cosets
Summary
Since Zadeh [1] introduced the concept of a fuzzy set in 1965, various algebraic structures have been fuzzified. By Proposition 9, Aα0 is a subgroup of G, x = xyy−1 ∈ Aα0 It follows that A(x) ⩾ α0 > λ, which is a contradiction. Let G be a cyclic group with generators a and b, A a (λ, μ)-fuzzy subgroup of G. Let G be a cyclic group of a prime order with generator a, A a (λ, μ)-fuzzy subgroup of G. Let A be a (λ, μ)-fuzzy normal subgroup of G and x ∈ G. By Proposition 21, Aμ is a normal subgroup of G and yxy−1 ∈ Aμ. By Proposition 23, A is a (λ, μ)-fuzzy normal subgroup of G. If G is a cyclic group and A is a (λ, μ)-fuzzy subgroup of G, A is a (λ, μ)-fuzzy normal subgroup of G
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