nLab symmetric midpoint algebra

Idea

A symmetric midpoint algebra is an abstract treatment of the operation $a \vert b \coloneqq \frac{a + b}{2}$, which finds the midpoint between $a$ and $b$, including the ideas that the midpoint is independent of the order of $a$ and $b$.

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.

Related concepts

• midpoint algebra

• cancellative midpoint algebra

• symmetric cancellative midpoint algebra

• symmetric midpoint group

• dyadic rational algebra

References

• Marshall H Stone, Postulates for the barycentric calculus, Ann. Mat. Pura. Appl. (4), 29:25–30, 1949.

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