nLab
symmetric midpoint group

Contents

Contents

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 GG with an element 0:G0:G, a function :GG-: G \to G, a binary operation ()+():G×GG(-)+(-):G \times G \to G and a binary operation ()|():G×GG(-)\vert(-): G \times G \to G such that

Examples

The rational numbers, real numbers, and the complex numbers with a|ba+b2a \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.

Created on May 31, 2021 at 21:12:58. See the history of this page for a list of all contributions to it.