# nLab symmetric cancellative midpoint algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A symmetric midpoint algebra that is also a cancellative midpoint algebra.

## Definition

A symmetric cancellative midpoint algebra is a symmetric midpoint algebra $(M,\vert, \odot, (-)^\bullet)$ that satisfies the cancellative property:

• for all $a$, $b$, and $c$ in $M$, if $a \vert b = a \vert c$, then $b = c$

## Properties

For all $a$ and $b$ in $M$, $a = b$ if and only if $a^\bullet \vert b = \odot$.

## 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 cancellative midpoint algebras.

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

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