Let E be a set of inequalities between finite Σ-terms. Let V_ω and V_r denote the varieties of all ω-continuous ordered Σ-algebras and regular ordered Σ-algebras satisfying E, respectively. We prove that the free V_r-algebra R(X) on generators X is the subalgebra of the corresponding free V_ω-algebra F_ω(X) determined by those elements of F_ω(X) denoted by the regular Σ-coterms. We actually establish this fact as a special case of a more general construction for families of algebras specified by syntactically restricted completeness and continuity properties. Thus our result is also applicable to ordered regular algebras of higher order.