nLab totally ordered ring

Contents

Context

Algebra

(0,1)-Category theory

Contents

Idea

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

Definition

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

A totally ordered ring is an pseudolattice ordered ring RR such that for all elements aa in RR, a0a \leq 0 or a0-a \leq 0.

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

Examples

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

References

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

Created on June 18, 2021 at 17:39:46. See the history of this page for a list of all contributions to it.