nLab symmetric closed midpoint algebra

Definition

A symmetric closed midpoint algebra is a closed midpoint algebra $(M,\vert)$ with a function $(-)^{\bullet}: M \to M$ such that:

• for all $a$ and $b$ in $M$, $a^{\bullet} \vert a = \odot$, where $odot = \bot\vert\top$

Properties

• $\bot=\top^{\bullet}$

• for all $a$ in $M$, $(a^{\bullet})^{\bullet} = a$

• for all $a$ and $b$ in $M$, $(a \vert b)^{\bullet} = a^{\bullet} \vert b^{\bullet}$

Therefore: A symmetric closed midpoint algebra is the same thing as a symmetric cancellative midpoint algebra $(M,\vert, \odot, (-)^\bullet)$ with two elements $\bot:M$ and $\top:M$ such that $\bot\vert\top = \odot$.

Proposition

The initial object among symmetric closed midpoint algebras is the unit interval in the dyadic rational numbers.

Related concepts

• definite integration

• 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, pdf]