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It is well known that classical varieties of $\Sigma$-algebras correspondbijectively to finitary monads on $\mathsf{Set}$. We present an analogousresult for varieties of ordered $\Sigma$-algebras, i.e., classes presented byinequations between $\Sigma$-terms. We prove that they correspond bijectivelyto strongly finitary monads on $\mathsf{Pos}$. That is, those finitary monadswhich preserve reflexive coinserters. We deduce that strongly finitary monadshave a coinserter presentation, analogous to the coequaliser presentation offinitary monads due to Kelly and Power. We also show that these monads areliftings of finitary monads on $\mathsf{Set}$.

A categorical view of varieties of ordered algebras