Partially Ordered Ring

In abstract algebra, a partially ordered ring is a ring (''A'', +, ·), together with a ''compatible partial order'', that is, a partial order $\,\leq\,$ on the underlying set ''A'' that is compatible with the ring operations in the sense that it satisfies:
$x \leq y \text x + z \leq y + z$
and
$0 \leq x \text 0 \leq y \text 0 \leq x \cdot y$
for all $x, y, z\in A$. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring $(A, \leq)$ where partially ordered additive group is Archimedean.

An ordered ring, also called a totally ordered ring, is a partially ordered ring $(A, \leq)$ where $\,\leq\,$ is additionally a total order.

An l-ring, or lattice-ordered ring, is a partially ordered ring $(A, \leq)$ where $\,\leq\,$ is additionally a lattice order.

Properties

The additive group of a partially ordered ring is always a partially ordered group.

The set of non-negative elements of a partially ordered ring (the set of elements $x$ for which $0 \leq x,$ also called the positive cone of the ring) is closed under addition and multiplication, that is, if $P$ is the set of non-negative elements of a partially ordered ring, then $P + P \subseteq P$ and $P \cdot P \subseteq P.$ Furthermore, $P \cap (-P) = \.$

The mapping of the compatible partial order on a ring $A$ to the set of its non-negative elements is one-to-one; that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists.

If $S \subseteq A$ is a subset of a ring $A,$ and:
# $0 \in S$
# $S \cap (-S) = \$
# $S + S \subseteq S$
# $S \cdot S \subseteq S$
then the relation $\,\leq\,$ where $x \leq y$ if and only if $y - x \in S$ defines a compatible partial order on $A$ (that is, $(A, \leq)$ is a partially ordered ring).

In any l-ring, the $, x,$ of an element $x$ can be defined to be $x \vee(-x),$ where $x \vee y$ denotes the maximal element. For any $x$ and $y,$
$, x \cdot y, \leq , x, \cdot , y,$
holds.

f-rings

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring $(A, \leq)$ in which $x \wedge y = 0$ and $0 \leq z$ imply that $zx \wedge y = xz \wedge y = 0$ for all $x, y, z \in A.$ They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled "Lattice-ordered rings", in an attempt to restrict the class of l-rings so as to eliminate a number of pathological examples. For example, Birkhoff and Pierce demonstrated an l-ring with 1 in which 1 is not positive, even though it is a square. The additional hypothesis required of f-rings eliminates this possibility.

Example

Let $X$ be a Hausdorff space, and $\mathcal(X)$ be the space of all continuous, real-valued functions on $X.$ $\mathcal(X)$ is an Archimedean f-ring with 1 under the following pointwise operations:
+ gx) = f(x) + g(x)
gx) = f(x) \cdot g(x)
\wedge gx) = f(x) \wedge g(x).

From an algebraic point of view the rings $\mathcal(X)$
are fairly rigid. For example, localisations, residue rings or limits of rings of the form $\mathcal(X)$ are not of this form in general. A much more flexible class of f-rings containing all rings of continuous functions and resembling many of the properties of these rings is the class of real closed rings.

Properties

* A direct product of f-rings is an f-ring, an l-subring of an f-ring is an f-ring, and an l-homomorphic image of an f-ring is an f-ring.

* $, xy, = , x, , y,$ in an f-ring.

* The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.

* Every ordered ring is an f-ring, so every sub-direct union of ordered rings is also an f-ring. Assuming the axiom of choice, a theorem of Birkhoff shows the converse, and that an l-ring is an f-ring if and only if it is l-isomorphic to a sub-direct union of ordered rings. Some mathematicians take this to be the definition of an f-ring.

Formally verified results for commutative ordered rings

IsarMathLib, a library for the Isabelle theorem prover, has formal verifications of a few fundamental results on commutative ordered rings. The results are proved in the ring1 context.

Suppose $(A, \leq)$ is a commutative ordered ring, and $x, y, z \in A.$ Then:

See also

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References

Further reading

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* Gillman, Leonard; Jerison, Meyer Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp

External links

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* {{PlanetMath, title = Partially Ordered Ring , urlname = PartiallyOrderedRing

Ring theory
Ordered algebraic structures