Some studies in the approximation of $$(\in _{\gamma }, \in _{\gamma }\vee q_{\delta })$$-fuzzy substructures in quantales

Abstract

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, was used for the first time by Mulvey in 1986. In many applied disciplines like theoretical computer science, algebraic theory, rough set theory, topological theory and linear logic, the use of fuzzified algebraic structures specially quantales plays an important role. In the present paper, the concept of generalized roughness for $$\left( \in _{\gamma },\in _{\gamma }\vee q_{\delta }\right) $$-fuzzy filters in quantales, is introduced. The concept is extended to the approximations of $$\left( \in _{\gamma },\in _{\gamma }\vee q_{\delta }\right) $$-fuzzy ideals and $$\left( \in _{\gamma },\in _{\gamma }\vee q_{\delta }\right) $$-fuzzy subquantales. Moreover, this concept is applied to study the approximations of $$\left( \in _{\gamma },\in _{\gamma }\vee q_{\delta }\right) $$-fuzzy prime ideals and $$\left( \in _{\gamma },\in _{\gamma }\vee q_{\delta }\right) $$-fuzzy semi-prime ideals.