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

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

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

If the underlying poset of a 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 $(a_{ij})\le (b_{ij})$ whenever $a_{ij}\le b_{ij}$ for all $i,j$ .

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

Bibliography

1

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