Finitary Monads on the Category of Posets

TL;DR

This paper characterizes finitary monads on the category of posets as free-algebra monads of algebraic varieties defined by inequations, extending to enriched monads with coherent algebra structures.

Contribution

It provides a new characterization of finitary monads on Pos and extends the theory to enriched monads with monotone operations.

Findings

Finitary monads on Pos correspond to free-algebra monads of algebraic varieties.

Finitary enriched monads on Pos are characterized by coherent algebra varieties.

The work links algebraic inequations to monad structures on posets.

Abstract

Finitary monads on $\mathsf{Pos}$ are characterized as the precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analagously, finitary enriched monads on $\mathsf{Pos}$ are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.