Partially ordered ring

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

$x\leq y$ implies $x + z\leq y + z$

and

$0\leq x$ and $0\leq y$ imply that $0\leq x\cdot y$

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

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

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, i.e., 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;^[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 S 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 $x\leq y$ iff $y - x\in S$ defines a compatible partial order on A (ie. $(A, \leq)$ is a partially ordered ring).^[2]

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.^[3]

f-rings

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring $(A, \leq)$ in which $x\wedge y = 0$^[4] 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 negative, even though being a square.^[2] 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 point-wise operations:

$[f + g](x) = f(x) + g(x)$

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

$[f\wedge g](x) = f(x)\wedge g(x).$^[2]

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.^[3]

$|xy|=|x||y|$ in an f-ring.^[3]

The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.^[5]

Every ordered ring is an f-ring, so every subdirect 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 subdirect union of ordered rings.^[2] Some mathematicians take this to be the definition of an f-ring.^[3]

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 $(A, \leq)$ 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
$x\leq y$ iff $x - y\leq 0$	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 iff A has no zero divisors.	OrdRing_ZF_3_L3
If A is non-trivial ($0\neq 1$), then it is infinite.	ord_ring_infinite

References

1. Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics 17: 434–448. doi:10.4153/cjm-1965-044-7.
2. Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica 104 (3–4): 163–215. doi:10.1007/BF02546389.
3. Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez. Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8.
4. $\wedge$ denotes infimum.
5. Hager, Anthony W.; Jorge Martinez (2002). "Functorial rings of quotients—III: The maximum in Archimedean f-rings". Journal of Pure and Applied Algebra 169: 51–69. doi:10.1016/S0022-4049(01)00060-3.
6. "IsarMathLib" (PDF). Retrieved 2009-03-31.

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, Partially Ordered Ring". Encyclopedia of Mathematics. Retrieved 2009-04-03.
• "Partially Ordered Ring". PlanetMath. Retrieved 2009-03-30.