We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets.
We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense).
A. Kurz and J. Velebil, “Quasivarieties and varieties of ordered algebras: regularity and exactness,” Mathematical Structures in Computer Science, vol. 27, no. 07, pp. 1153–1194, Oct. 2017. DOI: 10.1017/S096012951500050X
Cambridge University Press