Definitions Of Vague Ideal And Vague Prime Ideal Of A Γ -semiring

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Vague Prime Ideals of a Γ-Semirings-II Y.Bhargavi Research Scholar, Department of Mathematics, K.L.University, Guntur, India. yellabhargavi@gmail.com T.Eswarlal, Associate Professor, Department of Mathematics, K.L.University, Guntur, India. eswarlal@kluniversity.in
Abstract The concepts of vague ideal and vague prime ideal of a Γ-semiring which is characterized by a truth-membership function and a false membership function in a complete lattice satisfying inﬁnite meet distributive law (i.e., CompleteBrouwerianLattice)areintroduced. Furtherallvagueprimeideals of a Γ-semiring are determined by establishing a one-to-one correspondence between vague prime ideals of a Γ-semiring and the pair (I,α), where I is aprimeidealofaΓ-semiringand α isaprimeelementinthecompletelattice.
Key Words: Complete lattice, Brouwerian lattice, vague set, vague ideal, vague prime ideal.
Mathematics Subject Classiﬁcation: 08A72, 20N25, 03E72.
1.Introduction In 1965, Zadeh.L.A[16] introduced the study of fuzzy sets. Mathematically a fuzzy set on a set X is a mapping µ into [0, 1] of real numbers; for x in X, µ(x) is called the membership of x belonging to X. The membership function gives only an approximation for belonging but it does not give any information of not belonging. To overcome this, Gau.W.L and Buehrer.D.J[7] introduced the concept of vague sets. A vague set A of a set X is a pair of functions (tA,fA), where tA and fA are fuzzy sets on X satisfying tA(x)+fA(x)≤1, ∀ x ∈ X. A fuzzy set tA of X