Generalized Plonka Sums and Products

TL;DR

This paper provides a categorical framework for Plonka sums and products, extending their applicability to categories of algebras of semi-analytic and analytic monads with generalized arities.

Contribution

It introduces a unified categorical approach to Plonka sums and products using lax and oplax monad morphisms, broadening their scope to semi-analytic and analytic monads.

Findings

Generalized operations on algebras of semi-analytic monads

Extended arities to categories of regular and linear polynomials

Defined dual Plonka product operations on Kleisli categories

Abstract

We give an abstract categorical treatment of Plonka sums and products using lax and oplax morphisms of monads. Plonka sums were originally defined as operations on algebras of regular theories. Their arities are sup-semilattices. It turns out that even more general operations are available on the categories of algebras of semi-analytic monads. Their arities are the categories of the regular polynomials over any sup-semilattice, i.e. any algebra for the terminal semi-analytic monad. We also show that similar operations can be defined on any category of algebras of any analytic monad. This time we can allow the arities to be the categories of linear polynomials over any commutative monoid, i.e. any algebra for the terminal analytic monad. There are also dual operations of Plonka products. They can be defined on Kleisli categories of commutative monads.