# nLab symmetric midpoint algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

The idea of a symmetric midpoint algebra comes from Peter Freyd.

## Definition

A symmetric midpoint algebra is a midpoint algebra $(M,\vert)$ with an element $\odot:M$ and a function $(-)^{\bullet}: M \to M$ such that

• for all $a$ in $M$, $(a^{\bullet})^{\bullet} = a$

• for all $a$ and $b$ in $M$, $a^{\bullet} \vert a = \odot$

• for all $a$ and $b$ in $M$, $(a \vert b)^{\bullet} = a^{\bullet} \vert b^{\bullet}$

## Properties

$\odot$ is the only element in $M$ such that $\odot^\bullet = \odot$.

## Examples

The rational numbers, real numbers, and the complex numbers with $a \vert b \coloneqq \frac{a + b}{2}$, $\odot = 0$, and $a^{\bullet} = -a$ are examples of symmetric midpoint algebras.

The trivial group with $a \vert b = a \cdot b$, $\odot = 1$ and $a^{\bullet} = a^{-1}$ is a symmetric midpoint algebra.

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)