partially ordered ring

A ring R that is a poset at the same time is called a partially ordered ring, or a po-ring, if, for a,b,cR,

  • ab implies a+cb+c, and

  • 0a and 0b implies 0ab.

Note that R does not have to be associative.

If the underlying poset of a po-ring R is in fact a latticeMathworldPlanetmath, then R is called a lattice-ordered ring, or an l-ring for short.

Remark. The underlying abelian groupMathworldPlanetmath of a po-ring (with addition being the binary operationMathworldPlanetmath) is a po-group. The same is true for l-rings.

Below are some examples of po-rings:

Remark. Let R be a po-ring. The set R+:={rR0r} is called the positive cone of R.


Title partially ordered ring
Canonical name PartiallyOrderedRing
Date of creation 2013-03-22 16:55:04
Last modified on 2013-03-22 16:55:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 13J25
Classification msc 16W80
Classification msc 06F25
Synonym po-ring
Synonym l-ring
Synonym lattice-ordered ring
Defines lattice ordered ring
Defines positive cone