partially ordered ring
A ring that is a poset at the same time is called a partially ordered ring, or a po-ring, if, for ,
implies , and
and implies .
Note that does not have to be associative.
If the underlying poset of a po-ring is in fact a lattice, then is called a lattice-ordered ring, or an l-ring for short.
Below are some examples of po-rings:
Clearly, any (totally) ordered ring is a po-ring.
Any matrix ring over an ordered field is an l-ring if we define whenever for all .
Remark. Let be a po-ring. The set is called the positive cone of .
- 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
|Title||partially ordered ring|
|Date of creation||2013-03-22 16:55:04|
|Last modified on||2013-03-22 16:55:04|
|Last modified by||CWoo (3771)|
|Defines||lattice ordered ring|