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 $a,b,c\\in R$ ,

•
$a\\leq b$ implies $a+c\\leq b+c$ , and
•
$0\\leq a$ and $0\\leq b$ implies $0\\leq ab$ .

Note that $R$ does not have to be associative.

If the underlying poset of a po-ring $R$ is in fact a lattice, then $R$ 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:

•
Clearly, any (totally) ordered ring is a po-ring.
•
The ring of continuous functions over a topological space is an l-ring.
•
Any matrix ring over an ordered field is an l-ring if we define $(a_{ij})\\leq(b_{ij})$ whenever $a_{ij}\\leq b_{ij}$ for all $i, j$ .

Remark. Let $R$ be a po-ring. The set $R^{+}:=\\{r\\in R\\mid 0\\leq r\\}$ is called the positive cone of $R$ .

References

1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).

单词	PartiallyOrderedRing
释义	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 $a,b,c\\in R$ , • $a\\leq b$ implies $a+c\\leq b+c$ , and • $0\\leq a$ and $0\\leq b$ implies $0\\leq ab$ . Note that $R$ does not have to be associative. If the underlying poset of a po-ring $R$ is in fact a lattice, then $R$ 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: • Clearly, any (totally) ordered ring is a po-ring. • The ring of continuous functions over a topological space is an l-ring. • Any matrix ring over an ordered field is an l-ring if we define $(a_{ij})\\leq(b_{ij})$ whenever $a_{ij}\\leq b_{ij}$ for all $i, j$ . Remark. Let $R$ be a po-ring. The set $R^{+}:=\\{r\\in R\\mid 0\\leq r\\}$ is called the positive cone of $R$ . References 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
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