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 $\leq$ 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:

$0\in S$
$S\cap(-S) = \{0\}$
$S + S\subseteq S$
$S\cdot S\subseteq S$

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 2 3 Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7.
1 2 3 4 5 6 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.
1 2 3 4 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.
↑ $\wedge$ denotes infimum.
↑ 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.
↑ "IsarMathLib" (PDF). Retrieved 2009-03-31.

External links

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

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Partially ordered ring

Properties

f-rings

Example

Properties

Formally verified results for commutative ordered rings

References

Further reading

External links