nLab cancellative midpoint algebra

Idea

The idea of cancellative midpoint algebras is due to Freyd 2008.

Definition

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

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

Examples

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

In general, if $(M,\vert)$ forms a quasigroup and not just a cancellative magma, then a unique $Z[1/2]$-module characterizes it. (proof)

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

Escardó & Simpson 2001 describe examples of this algebraic structure over many other categories besides Set.

Related concepts

• midpoint algebra

• symmetric cancellative midpoint algebra

• closed midpoint algebra

References

• Peter Freyd, Algebraic real analysis, Theory and Applications of Categories 20 10 (2008) 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, IEEE Computer Society (2001) 115–125

  [doi:10.1109/LICS.2001.932488, pdf]