nLab symmetric closed midpoint algebra

Contents

Contents

Idea

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

Definition

A symmetric closed midpoint algebra is a symmetric cancellative midpoint algebra (M,|,,() )(M,\vert, \odot, (-)^\bullet) with two elements :M\bot:M and :M\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|ba+b2a \vert b \coloneqq \frac{a + b}{2}, =12\odot = \frac{1}{2}, a =1aa^\bullet = 1 - a, =0\bot = 0, and =1\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.

References

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

Last revised on June 19, 2021 at 01:09:37. See the history of this page for a list of all contributions to it.