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symmetric cancellative midpoint algebra

Contents

Contents

Idea

A symmetric midpoint algebra that is also a cancellative midpoint algebra.

Definition

A symmetric cancellative midpoint algebra is a symmetric midpoint algebra (M,|,,() )(M,\vert, \odot, (-)^\bullet) 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

Properties

For all aa and bb in MM, a=ba = b if and only if a |b=a^\bullet \vert b = \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 cancellative 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 cancellative 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 1, 2021 at 16:13:55. See the history of this page for a list of all contributions to it.