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 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 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 Set . Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on Set to strongly finitary monads on Pos .