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Fraser and Horn, and independently Hu, studied varietiesof algebras satisfying the property that for everyevery congruenceis a product congruence, i.e.,for somei  = 1, 2. Varieties of rings with identity and congruence distributive varieties of algebras satisfy this property. It is easy to show that an algebrahas the Fraser–Horn–Hu property if and only if the mapis a lattice isomorphism fromtoThe property was later referred to in the literature as the Fraser–Horn–Hu property. It turns out that the Fraser–Horn–Hu property is a Mal’cev condition for varieties. In this paper, we generalize this property to varieties of ordered algebras. The classic result of Fraser, Horn and Hu follows as a special case.

Fraser–Horn–Hu property for ordered algebras