Here a variety of ordered (finitary) algebras is defined as a category of algebras presented by inequations between terms. In [A. Kurz and J. Velebil, Math. Struct. Comput. Sci. 27, No. 7, 1153–1194 (2015; Zbl 1423.08007)] (the source being incorrectly referred in the paper as Logical Methods in Computer Science), Kurz and Velebil proved that they correspond (by dual equivalence) to strongly finitary monads on the category \(\textbf{POS}\) of posets. In the paper’s context, strongly finitary means that the endofunctor on \(\textbf{POS}\) is enriched (i.e., locally monotone) and preserves (weighted) sifted colimits (or, equivalently, filtered colimits and coinserters of reflexive pairs). The authors give a “much simpler” proof of this result, noting also that a “completely different proof has been presented, after the submission of our paper, by Rosický (2012)” (The mentioned year “2012” appears to be a misprint for what is actually [J. Rosický, “Metric monads”, Preprint, arXiv:2012.14641]). From the Abstract, the authors emphasize that hence “strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power.” It adds that “We also show that these monads are liftings of finitary monads on \(\textbf{Set}\). Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on \(\textbf{Set}\) to strongly finitary monads on \(\textbf{POS}\).”
For a more recent intrinsic characterization of varieties of ordered algebras as (\(\textbf{POS}\)-enriched) categories – in Lawvere’s style –, see [J. Adámek and J. Rosický, “Varieties of ordered algebras as categories”, Preprint, arXiv:2110.06613].