# Birings and plethories of integer-valued polynomials

Jesse Elliott^{[1]}

- [1] Department of Mathematics California State University, Channel Islands Camarillo, California 93012

Actes des rencontres du CIRM (2010)

- Volume: 2, Issue: 2, page 53-58
- ISSN: 2105-0597

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topElliott, Jesse. "Birings and plethories of integer-valued polynomials." Actes des rencontres du CIRM 2.2 (2010): 53-58. <http://eudml.org/doc/196274>.

@article{Elliott2010,

abstract = {Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor $\operatorname\{Hom\}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-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 $\{\operatorname\{Int\}\}(D)$ has such a structure if $D = A$ is a domain such that the natural $D$-algebra homomorphism $\theta _n: \{\bigotimes _D\}_\{i = 1\}^n \{\operatorname\{Int\}\}(D) \rightarrow \{\operatorname\{Int\}\}(D^n)$ is an isomorphism for $n = 2$ and injective for $n \le 4$. 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 $\operatorname\{Hom\}_D(\{\operatorname\{Int\}\}(D),-)$ 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.},

affiliation = {Department of Mathematics California State University, Channel Islands Camarillo, California 93012},

author = {Elliott, Jesse},

journal = {Actes des rencontres du CIRM},

keywords = {Biring; plethory; integer-valued polynomial},

language = {eng},

number = {2},

pages = {53-58},

publisher = {CIRM},

title = {Birings and plethories of integer-valued polynomials},

url = {http://eudml.org/doc/196274},

volume = {2},

year = {2010},

}

TY - JOUR

AU - Elliott, Jesse

TI - Birings and plethories of integer-valued polynomials

JO - Actes des rencontres du CIRM

PY - 2010

PB - CIRM

VL - 2

IS - 2

SP - 53

EP - 58

AB - Let $A$ and $B$ be commutative rings with identity. An $A$-$B$-biring is an $A$-algebra $S$ together with a lift of the functor $\operatorname{Hom}_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An $A$-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 ${\operatorname{Int}}(D)$ has such a structure if $D = A$ is a domain such that the natural $D$-algebra homomorphism $\theta _n: {\bigotimes _D}_{i = 1}^n {\operatorname{Int}}(D) \rightarrow {\operatorname{Int}}(D^n)$ is an isomorphism for $n = 2$ and injective for $n \le 4$. 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 $\operatorname{Hom}_D({\operatorname{Int}}(D),-)$ 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.

LA - eng

KW - Biring; plethory; integer-valued polynomial

UR - http://eudml.org/doc/196274

ER -

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