Maps Given by Iterated Operations: Partial Iterates, Images of Natural Numbers, Powers and Operations over a Finite Set

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let maps and their values be as in Sets and Maps: Ordinary Notation §maps, and binary operations as in Binary Operations on a Set §notation.

¶ Let $\ast$ be a binary operation on a set $X$ and let $a:\mathbb{N}\to X$, $k\mapsto a_{k}$. There is exactly one map $s:\mathbb{N}\to X$ with $s(n)=\mathop{\ast}\limits_{k=1}^{n}a_{k}$ for every $n\in\mathbb{N}$, the iterated operation being as in Iterated Operations: Finite Sums and Finite Products §restriction.

¶ Let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring, and for $n\in\mathbb{N}_{0}$ let $n_{R}$ be the image of $n$ in $R$. There is exactly one map $\nu:\mathbb{N}_{0}\to R$ with $\nu(n)=n_{R}$ for every $n\in\mathbb{N}_{0}$.

¶ Let $\cdot$ be a binary operation on a set $X$ that has a neutral element, let $x\in X$, and for $n\in\mathbb{N}_{0}$ let $x^{n}$ be as in Powers with Exponents in the Natural Numbers with Zero §power and Powers with Exponents in the Natural Numbers with Zero §zero. There is exactly one map $p:\mathbb{N}_{0}\to X$ with $p(n)=x^{n}$ for every $n\in\mathbb{N}_{0}$.

¶ Let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring and let $x\in R$. Then $1$ is a neutral element of $\cdot$, so $x^{n}$ is defined for $n\in\mathbb{N}_{0}$, for the multiplication $\cdot$, by Powers with Exponents in the Natural Numbers with Zero §power and Powers with Exponents in the Natural Numbers with Zero §zero; $x^{0}=1$, and there is exactly one map $p:\mathbb{N}_{0}\to R$ with $p(n)=x^{n}$ for every $n\in\mathbb{N}_{0}$.

¶ Let $\ast$ be an associative and commutative binary operation on a set $X$, let $A$ be a set and $B$ a finite set such that $B\neq\emptyset$ or $\ast$ has a neutral element, and let $f:A\times B\to X$. For $x\in A$ let $f_{x}$ be the map $B\to X$, $y\mapsto f(x,y)$, of Sets and Maps: Ordinary Notation §maps, and let $\mathop{\ast}\limits_{y\in B}f_{x}(y)$ be the iterated operation over $B$ of Sums and Products over a Finite Set and over an Interval §operation and Sums and Products over a Finite Set and over an Interval §empty. There is exactly one map $h:A\to X$ with $h(x)=\mathop{\ast}\limits_{y\in B}f_{x}(y)$ for every $x\in A$.