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:

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: $$[f+g](x)=f(x)+g(x)$$ $$[fg](x)=f(x)\cdot g(x)$$ $$[f\wedge g](x)=f(x)\wedge g(x).$$

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 ring1 context.

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

	by
The additive group of $A$ is an ordered group	OrdRing_ZF_1_L4
$x\leq y{\text{ if and only if }}x-y\leq 0$	OrdRing_ZF_1_L7
$x\leq y$ and $0\leq z$ imply $xz\leq yz$ and $zx\leq zy$	OrdRing_ZF_1_L9
$0\leq 1$	ordring_one_is_nonneg
$|xy|=|x||y|$	OrdRing_ZF_2_L5
$|x+y|\leq |x|+|y|$	ord_ring_triangle_ineq
$x$ 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 $(A,\leq )$ is closed under multiplication if and only if $A$ has no zero divisors.	OrdRing_ZF_3_L3
If $A$ is non-trivial ($0\neq 1$), 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

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.