Categories which are varieties of classical or ordered algebras

Following ideas of Lawvere and Linton we prove that classical varieties are precisely the exact categories with a varietal generator. This means a strong generator which is abstractly finite and regularly projective. An analogous characterization of varieties of ordered algebras is also presented. We work with order-enriched categories, and introduce the concept of subcongruence (cor- responding to congruence in ordinary categories): it is a relation which is order-reflexive and transitive. Varieties of ordered algebras are precisely the categories with effective subcongruences and a subvarietal generator. This means a strong generator which is abstractly finite and subregularly projective.