Varieties of ordered algebras

摘要

A variety of ordered algebras is a class K of ordered algebras satisfying satisfying a set of inequalities t≤t′. It is shown that a class K of ordered algebras is a variety if K is closed under subalgebras, products, and certain homomorphic images. The process of obtaining a “canonical” ω-completion of an ordered algebra is analyzed and it is shown that varieties of ordered algebras are closed with respect to ω-completion.The concluding sections concern (i) a connection between ordered algebras and ordered algebraic theories, and (ii) a logic of inequalities, analogous to equational logic. A completeness theorem for this logic is proved.