Generalized roughness in fuzzy filters and fuzzy ideals with thresholds in ordered semigroups

In the present paper, concept of roughness for fuzzy filters with thresholds \(\left( u_{1},u_{2}\right) \) in ordered semigroups is introduced. Then, this concept is extended to fuzzy bi-filters with thresholds and fuzzy quasi-filters with thresholds. Further approximations of fuzzy ideals with thresholds, fuzzy bi-ideals with thresholds, and fuzzy interior ideals with thresholds are studied. Moreover, this concept is applied to study approximations of fuzzy quasi-ideals with thresholds and semiprime fuzzy quasi-ideals with thresholds.