nLab symmetric closed midpoint algebra

Idea

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

Definition

A symmetric closed midpoint algebra is a symmetric cancellative midpoint algebra $(M,\vert, \odot, (-)^\bullet)$ with two elements $\bot:M$ and $\top:M$ such that $\bot\vert\top = \odot$.

Properties

Every symmetric closed midpoint algebra with $\bot = \top$ is trivial.

Examples

The unit interval with $a \vert b \coloneqq \frac{a + b}{2}$, $\odot = \frac{1}{2}$, $a^\bullet = 1 - a$, $\bot = 0$, and $\top = 1$ is an example of a symmetric closed midpoint algebra.

The set of truth values in Girard’s linear logic is a symmetric closed midpoint algebra.

Related concepts

• closed midpoint algebra

• symmetric cancellative midpoint algebra

• minor scale

• dyadic rational module

References

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