Partially ordered ring

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) = {0}.

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:

1. 0 \in S
2. S \cap (-S) = {0}
3. S + S \subseteq S
4. 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 absolute value |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{C}(X) be the space of all continuous, real-valued functions on X. \mathcal{C}(X) is an Archimedean f-ring with 1 under the following pointwise operations: f + g = f(x) + g(x) fg = f(x) \cdot g(x) f \wedge g = f(x) \wedge g(x).

From an algebraic point of view the rings \mathcal{C}(X) are fairly rigid. For example, localisations, residue rings or limits of rings of the form \mathcal{C}(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.

References

References

1. Anderson, F. W.. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics.
2. Johnson, D. G.. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica.
3. Henriksen, Melvin. (1997). "Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995". Kluwer Academic Publishers.
4. $\backslash wedge$ denotes [[infimum]].
5. Hager, Anthony W.. (2002). "Functorial rings of quotients—III: The maximum in Archimedean f-rings". Journal of Pure and Applied Algebra.