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 ,
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implies , and
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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.
Remark. The underlying abelian group of a po-ring (with addition being the binary operation) is a po-group. The same is true for l-rings.
Below are some examples of po-rings:
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Clearly, any (totally) ordered ring is a po-ring.
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The ring of continuous functions over a topological space is an l-ring.
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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 .
References
- 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
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 |