Birings and plethories of integer-valued polynomials
Jesse Elliott

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 and be commutative rings with identity. An {\it --biring} is an -algebra together with a lift of the functor from -algebras to sets to a functor from -algebras to -algebras. An {\it -plethory} is a monoid object in the monoidal category, equipped with the composition product, of --birings. The polynomial ring is an initial object in the category of such structures. The -algebra has such a structure if is a domain such that the natural -algebra homomorphism is an isomorphism for and injective for . This holds in particular if is an isomorphism for all , which in turn holds, for example, if is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor…
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Taxonomy
TopicsRings, Modules, and Algebras · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
