On Free ω-Continuous and Regular Ordered Algebras.

Let E be a set of inequalities between finite {Sigma}-terms. Let V_{omega} and V_r denote the varieties of all {omega}-continuous ordered {Sigma}-algebras and regular ordered {Sigma}-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_{omega}-algebra F_{omega}(X) determined by those elements of F_{omega}(X) denoted by the regular {Sigma}-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.