Levi-Civita field

Description

A function $a \colon \Q \to \R$ has left-finite support if for every $q \in \Q$ there are only finitely many $r < q$ with $a(r) \ne 0$. The Levi-Civita field $\mathcal{R}$ is the set of such functions, written as formal sums $a = \sum_{q \in \Q} a(q)\, \varepsilon^q$. Addition is termwise, and the product is $(ab)(q) = \sum_{r + s = q} a(r) b(s)$, a finite sum because both supports are left-finite. For $a \ne 0$, let $q_0$ be the least $q$ with $a(q) \ne 0$; then $a > 0$ means $a(q_0) > 0$. With these operations and this order, $\mathcal{R}$ is an ordered field that contains $\R$ as the constant series.