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Using the emph{belongs to} relation (in) and emph{quasi-coincidencewith} relation (q) between fuzzy points and fuzzy sets, the concept of(alpha, beta)-fuzzy subalgebras where alpha, , beta are any two of{in, {rm q}, in !vee , {rm q}, in !wedge , {rm q}} withalpha ne , in !wedge , {rm q} was introduced, and related propertieswere investigated in cite{BKMS42-703}. As a continuation of thepaper cite{BKMS42-703}, in this paper, the notion of a fuzzy subalgebra withthresholds is introduced, and its characterizations are obtained.Relations between a fuzzy subalgebra with thresholds and an(in, in! vee , {rm q})-fuzzy subalgebra are provided.