Varieties of ordered algebras as categories

A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere’s classical characterization of varieties of ordinary algebras. We also study the relationship of varieties to discrete Lawvere theories, and varieties as concrete categories over \(\mathbf{ Pos }\).