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. ↑ 1.0 1.1 1.2 Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics: 434–448.
  2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 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. ↑ 3.0 3.1 3.2 3.3 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". Retrieved 2009-03-31.

Further reading

External links