A categorical view of varieties of ordered algebras. (10th April 2022) - Adámek, J.; Dostál, M.; Velebil, J.
Abstract: It is well known that classical varieties of $Σ$ -algebras correspond bijectively to finitary monads on $\mathsf{Set}$ . We present an analogous result for varieties of ordered $Σ$ -algebras, that is, categories of algebras presented by inequations between $Σ$ -terms. We prove that they correspond bijectively to strongly finitary monads on $\mathsf{Pos}$ . That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power. We also show that these monads are liftings of finitary monads on $\mathsf{Set}$ . Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on $\mathsf{Set}$ to strongly finitary monads on $\mathsf{Pos}$ .