nLab midpoint algebra




The idea of a midpoint algebra comes from Peter Freyd.


A midpoint algebra is a magma (M,|)(M,\vert) that is commutative, idempotent, and medial:

  • for all aa in MM, a|a=aa \vert a = a

  • for all aa and bb in MM, a|b=b|aa \vert b = b \vert a

  • for all aa, bb, cc, and dd in MM, (a|b)|(c|d)=(a|c)|(b|d)(a \vert b) \vert (c \vert d) = (a \vert c) \vert (b \vert d)


The currying of the midpoint operation |\vert results in the contraction ()|:M(MM)(-)\vert : M \to (M \to M). Contractions are midpoint homomorphisms: for all aa, bb, and cc in MM, (a|)(b|c)=((a|)b)|((a|)c)(a \vert) (b \vert c) = ((a \vert) b) \vert ((a \vert) c).


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

The trivial group with a|b=aba \vert b = a \cdot b is a 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 06:55:11. See the history of this page for a list of all contributions to it.