nLab totally ordered ring

Idea

A totally ordered ring is an ordered ring whose order forms a total order.

This definition is adapted from Peter Freyd‘s definition of a totally ordered abelian group:

A totally ordered ring is an lattice-ordered ring $R$ such that for all elements $a$ in $R$ , $a \leq 0$ or $-a \leq 0$ .

In a totally ordered ring, the join is usually called the maximum, while the meet is usually called the minimum

If the relation $\leq$ is only a preorder, then the prelattice-ordered ring $R$ is said to be a totally preordered ring.

The integers, the rational numbers, and the real numbers are totally ordered rings.

Peter Freyd, Algebraic real analysis, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

Last revised on January 2, 2025 at 22:20:45. See the history of this page for a list of all contributions to it.