nLab midpoint algebra

Idea

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

Definition

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

• for all $a$ and $b$ in $M$, $a \vert b = b \vert a$

• for all $a$ in $M$, $a \vert a = a$

• for all $a$, $b$, $c$, and $d$ in $M$, $(a \vert b) \vert (c \vert d) = (a \vert c) \vert (b \vert d)$

Properties

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

Examples

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

The trivial group with $a \vert b = a \cdot b$ is a midpoint algebra.

Related concepts

• convex space

• commutative magma

• cancellative midpoint algebra

• symmetric midpoint algebra

References

• J Aczel. On mean values. Bull. AMS, vol. 54, pp. 392–400, 1948. Perhaps the first appearance of the theory.

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

• Reinhold Heckmann. Probabilistic domains. In: Trees in Algebra and Programming — CAAP’94. Lecture Notes in Computer Science, vol 787. Springer, 1994. doi:10.1007/BFb0017479

• 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)

• Martín Escardó and Alex Simpson, Euclidean interval objects in categories with finite products [arXiv:2504.21551]