[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] \section{Operads and n-ary functions} ... We begin with $n$-ary functions in sets, and then go on to multilinear functions in abelian groups, and finally to the same in modules. $$\mathrm{lcm}:\sum_{i:\mathrm{Fin}(n)} \mathrm{Fin}(n) \to \mathbb{N}$$ \section{Partitions} We need partitions of finite sets and symmetry in order to define finite sums and products on commutative rings. \section{Smooth real numbers} Let $\mathbb{R}[X]$ be the real polynomial ring generated by the natural numbers, with sequence of indeterminants $X:\mathbb{N} \to \mathbb{R}[X]$. We define the smooth real numbers to be the quotient ring $$\mathcal{R} \coloneqq \frac{\mathbb{R}[X]}{X(n)^{n + 2} \sim 0}$$ This implies that the smooth real numbers are a local ring, where the non-invertible elements are those whose real component is zero, and thus are called **purely infinitesimal real number**. Given natural number $n$ and lists of natural numbers $m:\mathrm{Fin}(n) \to \mathbb{N}$ and $p:\mathrm{Fin}(n) \to \mathbb{N}$, let $r$ be a finite product of powers of infintesimals $$r \coloneqq \prod_{i:\mathrm{Fin}(n)} X(m(i))^{p(i)}$$ The order to which the finite product of powers of infintesimals vanishes is the least common multiple of the natural numbers $(m(i) + 2) \div p(i)$ $$\mathrm{Order}(r) \coloneqq \lcm_{i:\mathrm{Fin}(n)} (m(i) + 2) \div p(i)$$ Every purely infinitesimal real number can be expressed as a linear combination of a finite product of powers of infintesimals. The order to which purely infinitesimal real number vanishes is at most the least common multiple of the order of all the finite product of powers of infintesimals used to express the purely infinitesimal real number. By definition, $\mathbb{R}$ is the reduced part of $\mathcal{R}$. category: redirected to nlab