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 ${\displaystyle \leq }$ on the underlying set A that is compatible with the ring operations in the sense that it satisfies:

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

and

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

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 x for which ${\displaystyle 0\leq x}$, 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 ${\displaystyle P+P\subseteq P}$ and ${\displaystyle P\cdot P\subseteq P}$. Furthermore, ${\displaystyle P\cap (-P)=\{0\}}$.

The mapping of the compatible partial order on a ring 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. ${\displaystyle 0\in S}$
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 ${\displaystyle x\leq y}$ iff ${\displaystyle y-x\in S}$ defines a compatible partial order on A (ie. ${\displaystyle (A,\leq )}$ is a partially ordered ring).[2]

In any l-ring, the absolute value ${\displaystyle |x|}$ of an element x can be defined to be ${\displaystyle x\vee (-x)}$, where ${\displaystyle x\vee y}$ denotes the maximal element. For any x and 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 negative, even though it is a square.[2] The additional hypothesis required of f-rings eliminates this possibility.

Example

Let X be a Hausdorff space, and ${\displaystyle {\mathcal {C}}(X)}$ be the space of all continuous, real-valued functions on 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.

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]

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

	by
The additive group of A is an ordered group	OrdRing_ZF_1_L4
${\displaystyle x\leq y}$ iff ${\displaystyle x-y\leq 0}$	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
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 iff A has no zero divisors.	OrdRing_ZF_3_L3
If A is non-trivial (${\displaystyle 0\neq 1}$), 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 (ed.). 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. ${\displaystyle \wedge }$ 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 2018-04-14.