Abstract:We introduce the notions of quasi-coincidence with relation between a fuzzy interval value and an interval valued fuzzy set, which is generalization quasi-coincidence with relation of ordinary fuzzy point and fuzzy set. With this objective in view, we introduce the notion of interval valued (∈,∈∨q)-fuzzy B-algebras, and discuss their basic properties and some equivalent descriptions. It’s pointed that an interval valued fuzzy B-algebra with thresholds is an important generalization of a fuzzy B-algebra with thresholds in B-algebras.