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: $$x \leq y \text{ implies } x + z \leq y + z$$ and $$0 \leq x \text{ and } 0 \leq y \text{ imply that } 0 \leq x \cdot y$$ 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 $A$'s 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, $$P \cap (-P) = \{0\}.$$

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: 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).
 * 1) $$0 \in S$$
 * 2) $$S \cap (-S) = \{0\}$$
 * 3) $$S + S \subseteq S$$
 * 4) $$S \cdot S \subseteq S$$

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 $$\mathcal{C}(X)$$ 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: $$[f + g](x) = f(x) + g(x)$$ $$[fg](x) = f(x) \cdot g(x)$$ $$[f \wedge g](x) = f(x) \wedge g(x).$$

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.

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  context.

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