Abstract:The concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set is considered. In fact, this is a generalization of quasi-coincidence of a fuzzy point with a fuzzy set. By using this new idea, the notions of interval valued (∈,∈∨q)-fuzzy subalgebras and (∈,∈∨)-fuzzy subalgebras of Q-algebras which are two generalizations of fuzzy subalgebras of Q-algebras, are introduced, and related properties are investigated. In particular, the concept of a fuzzy subgroup with thresholds is extended to the concept of an interval valued fuzzy subalgebra of Q-algebras with thresholds.