In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order ${\displaystyle \,\leq \,}$ on the underlying set A that is compatible with the ring operations in the sense that it satisfies:

$${\displaystyle x\leq y{\text{ implies }}x+z\leq y+z}$$

and

$${\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y}$$

for all ${\displaystyle x,y,z\in A}$.^[1] 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 ${\displaystyle (A,\leq )}$ where ${\displaystyle A}$'s partially ordered additive group is Archimedean.^[2]

An ordered ring, also called a totally ordered ring, is a partially ordered ring ${\displaystyle (A,\leq )}$ where ${\displaystyle \,\leq \,}$ is additionally a total order.^[1][2]

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

The mapping of the compatible partial order on a ring ${\displaystyle A}$ to the set of its non-negative elements is one-to-one;^[1] 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 ${\displaystyle S\subseteq A}$ is a subset of a ring ${\displaystyle A,}$ and:

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

then the relation ${\displaystyle \,\leq \,}$ where ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in S}$ defines a compatible partial order on ${\displaystyle A}$ (that is, ${\displaystyle (A,\leq )}$ is a partially ordered ring).^[2]

In any l-ring, the absolute value ${\displaystyle |x|}$ of an element ${\displaystyle x}$ can be defined to be ${\displaystyle x\vee (-x),}$ where ${\displaystyle x\vee y}$ denotes the maximal element. For any ${\displaystyle x}$ and ${\displaystyle y,}$

$${\displaystyle |x\cdot y|\leq |x|\cdot |y|}$$

holds.^[3]

f-rings

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring ${\displaystyle (A,\leq )}$ in which ${\displaystyle x\wedge y=0}$^[4] and ${\displaystyle 0\leq z}$ imply that ${\displaystyle zx\wedge y=xz\wedge y=0}$ for all ${\displaystyle x,y,z\in A.}$ 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.^[2] The additional hypothesis required of f-rings eliminates this possibility.

Example

Let ${\displaystyle X}$ be a Hausdorff space, and ${\displaystyle {\mathcal {C}}(X)}$ be the space of all continuous, real-valued functions on ${\displaystyle X.}$ ${\displaystyle {\mathcal {C}}(X)}$ is an Archimedean f-ring with 1 under the following pointwise operations:

$${\displaystyle [f+g](x)=f(x)+g(x)}$$

$${\displaystyle [fg](x)=f(x)\cdot g(x)}$$

$${\displaystyle [f\wedge g](x)=f(x)\wedge g(x).}$$

^[2]

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

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.^[6]

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

	by
The additive group of ${\displaystyle A}$ is an ordered group	OrdRing_ZF_1_L4
${\displaystyle x\leq y{\text{ if and only if }}x-y\leq 0}$	OrdRing_ZF_1_L7
${\displaystyle x\leq y}$ and ${\displaystyle 0\leq z}$ imply ${\displaystyle xz\leq yz}$ and ${\displaystyle zx\leq zy}$	OrdRing_ZF_1_L9
${\displaystyle 0\leq 1}$	ordring_one_is_nonneg
${\displaystyle |xy|=|x||y|}$	OrdRing_ZF_2_L5
${\displaystyle |x+y|\leq |x|+|y|}$	ord_ring_triangle_ineq
${\displaystyle x}$ is either in the positive set, equal to 0 or in minus the positive set.	OrdRing_ZF_3_L2
The set of positive elements of ${\displaystyle (A,\leq )}$ is closed under multiplication if and only if ${\displaystyle A}$ has no zero divisors.	OrdRing_ZF_3_L3
If ${\displaystyle A}$ is non-trivial (${\displaystyle 0\neq 1}$), then it is infinite.	ord_ring_infinite

See also

• Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
• Ordered field – Algebraic object with an ordered structure
• Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Ordered topological vector space
• Ordered vector space – Vector space with a partial order
• Partially ordered space – Partially ordered topological space
• Riesz space – Partially ordered vector space, ordered as a lattice

References

Further reading

• Birkhoff, G.; R. Pierce (1956). "Lattice-ordered rings". Anais da Academia Brasileira de Ciências. 28: 41–69.
• 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

• "Ordered ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Partially Ordered Ring at PlanetMath.