nLab
symmetric midpoint algebra

Contents

Contents

Idea

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

Definition

A symmetric midpoint algebra is a midpoint algebra (M,|)(M,\vert) with an element :M\odot:M and a function () :MM(-)^{\bullet}: M \to M such that

  • for all aa in MM, (a ) =a(a^{\bullet})^{\bullet} = a

  • for all aa and bb in MM, a |a=a^{\bullet} \vert a = \odot

  • for all aa and bb in MM, (a|b) =a |b (a \vert b)^{\bullet} = a^{\bullet} \vert b^{\bullet}

Properties

\odot is the only element in MM such that =\odot^\bullet = \odot.

Examples

The rational numbers, real numbers, and the complex numbers with a|ba+b2a \vert b \coloneqq \frac{a + b}{2}, =0\odot = 0, and a =aa^{\bullet} = -a are examples of symmetric midpoint algebras.

The trivial group with a|b=aba \vert b = a \cdot b, =1\odot = 1 and a =a 1a^{\bullet} = a^{-1} is a symmetric 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 18, 2021 at 20:50:16. See the history of this page for a list of all contributions to it.