nLab cancellative midpoint algebra




The idea of a cancellative midpoint algebra comes from Peter Freyd.


A cancellative midpoint algebra is a midpoint algebra (M,|)(M,\vert) that satisfies the cancellative property:

  • for all aa, bb, and cc in MM, if a|b=a|ca \vert b = a \vert c, then b=cb = c


The rational numbers, real numbers, and the complex numbers with a|ba+b2a \vert b \coloneqq \frac{a + b}{2} are examples of cancellative midpoint algebras.

The trivial group with a|b=aba \vert b = a \cdot b is a cancellative midpoint algebra.


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

  • Martín Escardó and Alex Simpson. A universal characterization of the closed Euclidean interval. In 16th Annual IEEE Symposium on Logic in Computer Science, Boston, Massachusetts, USA, June 16-19, 2001, Proceedings, pages 115–125. IEEE Computer Society, 2001. (doi:10.1109/LICS.2001.932488, pdf)

Last revised on December 26, 2023 at 06:55:56. See the history of this page for a list of all contributions to it.