\begin{abstract}
Let $F$ be the set of functions from an infinite set, $S$, to an ordered ring $R$. For $f$, $g$, and $h$ in $F$, the assertion $f = g + O(h)$ means that for some constant $C$, $|f(x) - g(x)| \leq C |h(x)|$ for every $x$ in $S$. Let $L$ be the first-order language with variables ranging over such functions, symbols for $0, +, -, \min, \max$, and absolute value, and a ternary relation $f = g + O(h)$. We show that the set of quantifier-free formulas in this language that are valid in the intended class of interpretations is decidable, and does not depend on the underlying set, $S$, or the ordered ring, $R$. If $R$ is a subfield of the real numbers, we can add a constant $1$ function, and, in fact, any sequence of functions with strictly increasing rates of growth, as well as multiplication by constants from any computable subfield.
\end{abstract}