Partially ordered ring explained

\leq on the underlying set A that is compatible with the ring operations in the sense that it satisfies:$$x\; \backslash leq\; y\; \backslash text\; x\; +\; z\; \backslash leq\; y\; +\; z$$and$$0\; \backslash leq\; x\; \backslash text\; 0\; \backslash leq\; y\; \backslash text\; 0\; \backslash leq\; x\; \backslash cdot\; y$$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 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.

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 ⋅ 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:0 \in S S \cap ( - S) = \{ 0 \} S + S \subseteq S S ⋅ 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

|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, $$|x\; \backslash cdot\; y|\; \backslash leq\; |x|\; \backslash cdot\; |y|$$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 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 l{C}(X) be the space of all continuous, real-valued functions on X. l{C}(X) is an Archimedean f-ring with 1 under the following pointwise operations:$$[f\; +\; g](x)\; =\; f(x)\; +\; g(x)$$$$[fg](x)\; =\; f(x)\; \backslash cdot\; g(x)$$$$[f\; \backslash wedge\; g](x)\; =\; f(x)\; \backslash wedge\; g(x).$$

From an algebraic point of view the rings

l{C}(X) are fairly rigid. For example, localisations, residue rings or limits of rings of the form l{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.^[5]
• 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.^[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 ifandonlyif 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 ≠ 1 ), then it is infinite.	ord_ring_infinite

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

Notes and References

1. Anderson . F. W. . Lattice-ordered rings of quotients . Canadian Journal of Mathematics . 434 - 448. 10.4153/cjm-1965-044-7 . 17.
2. Johnson . D. G. . December 1960 . A structure theory for a class of lattice-ordered rings . Acta Mathematica . 104 . 3 - 4 . 163 - 215 . 10.1007/BF02546389. free .
3. Book: Henriksen, Melvin . Melvin Henriksen

   . Melvin Henriksen . A survey of f-rings and some of their generalizations . 1 - 26 . Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23 - 30, 1995 . 1997 . W. Charles Holland and Jorge Martinez . 0-7923-4377-8 . Kluwer Academic Publishers . the Netherlands.

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. 10.1016/S0022-4049(01)00060-3. free .
6. Web site: IsarMathLib . 2009-03-31.