partially ordered ring
A ring $R$ that is a poset at the same time is called a partially ordered ring, or a poring, if, for $a,b,c\in R$,

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$a\le b$ implies $a+c\le b+c$, and

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$0\le a$ and $0\le b$ implies $0\le ab$.
Note that $R$ does not have to be associative.
If the underlying poset of a poring $R$ is in fact a lattice^{}, then $R$ is called a latticeordered ring, or an lring for short.
Remark. The underlying abelian group^{} of a poring (with addition being the binary operation^{}) is a pogroup. The same is true for lrings.
Below are some examples of porings:

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Clearly, any (totally) ordered ring is a poring.

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The ring of continuous functions over a topological space^{} is an lring.

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Any matrix ring over an ordered field is an lring if we define $({a}_{ij})\le ({b}_{ij})$ whenever ${a}_{ij}\le {b}_{ij}$ for all $i,j$.
Remark. Let $R$ be a poring. The set ${R}^{+}:=\{r\in R\mid 0\le r\}$ is called the positive cone of $R$.
References
 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
Title  partially ordered ring 
Canonical name  PartiallyOrderedRing 
Date of creation  20130322 16:55:04 
Last modified on  20130322 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  poring 
Synonym  lring 
Synonym  latticeordered ring 
Defines  lattice ordered ring 
Defines  positive cone 