Vague Gamma-Semirings

The concept of fuzzy set as opened up a new horizon in mathematical sciences. In fuzzy set theory, a fuzzy set ¿ is a class of objects X along with a grade of membership function. This membership function assigns to each object a grade of membership ranging between 0 and 1. This membership function gives only an approximation for belonging but it does not give any information of not belonging. To counter this problem, vague set was introduced in which it has both membership & non-membership mappings and it is a generalisation of fuzzy set. The object of thesis is, (i). The study of fuzzy ¿-semiring, fuzzy ideal and normal fuzzy ideal of a ¿-semiring. (ii). The study of vague ¿-semiring of a ¿-semiring. (iii). The study of vague ideals, operations on vague ideals and normal vague ideals of a ¿-semiring. (iv). The study of L-vague ideals, L-vague prime ideals and L-vague maximal ideals of a ¿-semiring, where L is a complete Brouwerian lattice. (v). The study of vague bi-ideals, vague quasi ideals and vague magnified translation of a vague set of a ¿-semiring. Finally we give an application of a vague magnified translation of a vague set in medical diagnosis.