# nLab periodic function

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

A periodic function on a $\mathbb{Z}$-module $M$ with a strict linear order $\lt$ where left and right addition are strictly monotonic is a function $f:M \to M$ with a positive element $a \in M$, $0 \lt a$ called the period, such that for all integers $n \in \mathbb{Z}$ and elements $x \in M$, $f(x) = f(x + n a)$, and for any other integer $b \in M$ where $0 \lt b$ and for all integers $n \in \mathbb{Z}$ and elements $x \in M$, $f(x) = f(x + n b)$, $a \leq b$.