PERMUTABLE AND MUTUALLY PERMUTABLE ON FUZZY GROUPS
by
Hassan Naraghi
Islamic Azad University, Ashtian Branch
Coauthors: Ali Iranmanesh and Hossein Naraghi

A fuzzy subset of a set X is mapping m: X →[0, 1]. The union and intersection of two fuzzy subset are defined using sup and inf point wise. We denote the set of all fuzzy subset of X by I^X. Let m ∈ I^X for a ∈ I, define m_a as follow:
m_a={x | x ∈ X, m(x) ≥ a}. m_a is called a-cut( or a-level) set of m.

Let m ∈ I^G. Then m is a fuzzy subgroup of G if and only if m_a is a subgroup of G, ∀a ∈ m(G)∪{b ∈ I | b ≤ m(e)}.

Let G be a group and let m and n be fuzzy subgroups of G.
(a) We say that m is permuted by n if for any a, b ∈ G, there exists x ∈ G such that m(x^-1ab) ≥ m(a), n(x) ≥ n(b).
(b) We say that m is permuted by n mutually if for any subgroup L of n_b that b ∈ Imn, we have been for any a ∈ G, l ∈ L, there exist l₁, l₂ of L such that m(l^-1₁al) ≥ m(a) and m(lal^-1₂) ≥ m(a).

Let G be a group and let m and n be fuzzy subgroups of G.
(a) We say m and n are permutable if m is permuted by n and n is permuted by m.
(b) We say m and n are mutually permutable if m is permuted by n mutually and n is permuted by m mutually.

Let m and n be fuzzy subgroups of G. If m and n are mutually permutable then m and nu are permutable.

Let m is a fuzzy normal subgroup of G. Then m permutes with every fuzzy subgroup of G mutually.

In this paper we will prove the following Theorems:

Theorem 1. Let m and n be fuzzy subgroups of G, then m and n are permutable if and if for any t ∈ Imm, s ∈ Imn, m_t, n_s are permutable.

Theorem 2 Let m and n be fuzzy subgroups of G. If m and n be permutable then mo n is a fuzzy subgroup of G.

Theorem 3 Let m and n be fuzzy subgroups of G, then m and n are mutually permutable if and if for any t ∈ Imm, s ∈ Imn, m_t, n_s are mutually permutable.

References

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John. N. Mordeson and D. S. Malik, `Fuzzy Commutative Algebra`, World Scientific publishing Co. Pte. Ltd. (1995).

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A.Rosenfeld, Fuzzy groups, J.Math.Anal.Appl. 35(1971), 512-517.

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L.A.Zadeh, Fuzzy sets, Inform.Control. 8(1965), 338-353.

Date received: March 5, 2008