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partially ordered ring
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(Definition)
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A ring  that is a poset at the same time is called a partially ordered ring, or a po-ring, if, for
 ,
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:
Remark. Let  be a po-ring. The set
 is called the positive cone of  .
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"partially ordered ring" is owned by CWoo.
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(view preamble)
| Other names: |
po-ring, l-ring, lattice-ordered ring |
| Also defines: |
lattice ordered ring, positive cone |
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Cross-references: ordered field, matrix ring, topological space, ring of continuous functions, ordered ring, po-group, binary operation, addition, abelian group, lattice, associative, implies, poset, ring
There are 3 references to this entry.
This is version 4 of partially ordered ring, born on 2007-04-12, modified 2007-05-07.
Object id is 9179, canonical name is PartiallyOrderedRing.
Accessed 1548 times total.
Classification:
| AMS MSC: | 06F25 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered rings, algebras, modules) | | | 16W80 (Associative rings and algebras :: Rings and algebras with additional structure :: Topological and ordered rings and modules) | | | 13J25 (Commutative rings and algebras :: Topological rings and modules :: Ordered rings) |
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Pending Errata and Addenda
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