A study on translational invariant vague set of a $$\Gamma $$-semiring

Abstract

In this paper, we introduce the concepts of translational invariant vague set of a $$\Gamma $$-semiring and units, associates, prime elements with respect to a vague set. Also, we define an ideals of a $$\Gamma $$-semiring generated by translational invariant vague set and an element and their properties are discussed. Further, we study the properties of homomorphic image and pre-image of translational invariant vague set under the $$\Gamma $$-semiring homomorphism and we prove that every homomorphic image of a prime ideal of a $$\Gamma $$-semiring generated by a $$\psi $$-prime element and translational invariant and f-invariant vague set is also a prime ideal.