nLab symmetric midpoint group

Idea

A symmetric midpoint algebra that is also a 0-truncated symmetric 2-group.

Definition

A symmetric midpoint group or abelian midpoint group is a set $G$ with an element $0:G$, a function $-: G \to G$, a binary operation $(-)+(-):G \times G \to G$ and a binary operation $(-)\vert(-): G \times G \to G$ such that

• $(G,+,0,-)$ is an abelian group

• $(G,\vert,0,-)$ is a symmetric midpoint algebra

• for all $a$ and $b$ in $G$, $a \vert b + a \vert b = a + b$

Examples

The rational numbers, real numbers, and the complex numbers with $a \vert b \coloneqq \frac{a + b}{2}$ are examples of symmetric midpoint groups.

The dyadic rational numbers are the free symmetric midpoint group on one generator.

The trivial group is a symmetric midpoint group, and is in fact a zero object in the category of symmetric midpoint groups.

Related concepts

• abelian group

• symmetric midpoint algebra

• dyadic rational number