  It is well known that classical varieties of $\Sigma$-algebras correspond
bijectively to finitary monads on $\mathsf{Set}$. We present an analogous
result for varieties of ordered $\Sigma$-algebras, i.e., classes presented by
inequations between $\Sigma$-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 coequaliser presentation of
finitary monads due to Kelly and Power. We also show that these monads are
liftings of finitary monads on $\mathsf{Set}$.
