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

6823 Words28 Pages
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 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 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