Integro-differential rings on species and derived structures

TL;DR

This paper develops an algebraic framework called integro-differential rings within the theory of species, unifying differential and integral operators and extending classical calculus concepts to combinatorial structures.

Contribution

It introduces the concept of integro-differential rings for species, connecting algebraic structures with calculus, and extends these ideas to virtual species and integral equations.

Findings

Integro-differential rings unify differential and integral operators in species theory.

The ring homomorphism from species to power series preserves the integro-differential structure.

Extensions to virtual species enable new algebraic and topological operations.

Abstract

In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are themselves an abstraction of classical calculus (incorporating its Fundamental Theorem). The results comprise (set) species as well as linear species. Localization of (set) species leads to the more general structure of modified integro-differential rings, previously employed in the algebraic treatment of Volterra integral equations. Furthermore, the ring homomorphism from species to power series via taking generating series is shown to be a (modified) integro-differential ring homomorphism. As an application, a topology and further algebraic operations are imported to virtual species from the general theory of integro-differential rings.