Birings and plethories of integer-valued polynomials

TL;DR

This paper explores the algebraic structures called birings and plethories related to integer-valued polynomials over certain rings, establishing conditions under which these structures exist and examining their properties.

Contribution

It introduces the concept of $A$-$B$-birings and $A$-plethories in the context of integer-valued polynomials, and identifies conditions for their existence over specific domains.

Findings

The structure of $Int(D)$ as an $A$-plethory is established under certain conditions.

The natural homomorphism $ heta_n$ is an isomorphism for all $n$ in specific classes of domains.

The functor $Hom_D(Int(D),-)$ is examined as a potentially new object in the theory of integer-valued polynomials.

Abstract

Let $A$ and $B$ be commutative rings with identity. An {\it $A$-$B$-biring} is an $A$-algebra $S$ together with a lift of the functor $Hom_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An {\it $A$-plethory} is a monoid object in the monoidal category, equipped with the composition product, of $A$-$A$-birings. The polynomial ring $A[X]$ is an initial object in the category of such structures. The $D$-algebra $Int(D)$ has such a structure if $D = A$ is a domain such that the natural $D$-algebra homomorphism $\theta_n: {\bigotimes_D}_{i = 1}^n Int(D) \longrightarrow Int(D^n)$ is an isomorphism for $n = 2$ and injective for $n \leq 4$. This holds in particular if $\theta_n$ is an isomorphism for all $n$, which in turn holds, for example, if $D$ is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor $Hom_D(Int(D),-)$ from $D$-algebras to $D$-algebras, which we hope to show is a new object worthy of investigation in the theory of integer-valued polynomials.