Partially ordered ring

Short description: Ring with a compatible partial order

In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order [math]\displaystyle{ \,\leq\, }[/math] on the underlying set A that is compatible with the ring operations in the sense that it satisfies: [math]\displaystyle{ x \leq y \text{ implies } x + z \leq y + z }[/math] and [math]\displaystyle{ 0 \leq x \text{ and } 0 \leq y \text{ imply that } 0 \leq x \cdot y }[/math] for all [math]\displaystyle{ x, y, z\in A }[/math]. 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 [math]\displaystyle{ (A, \leq) }[/math] where [math]\displaystyle{ A }[/math]'s partially ordered additive group is Archimedean.

An ordered ring, also called a totally ordered ring, is a partially ordered ring [math]\displaystyle{ (A, \leq) }[/math] where [math]\displaystyle{ \,\leq\, }[/math] is additionally a total order.

An l-ring, or lattice-ordered ring, is a partially ordered ring [math]\displaystyle{ (A, \leq) }[/math] where [math]\displaystyle{ \,\leq\, }[/math] 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 [math]\displaystyle{ x }[/math] for which [math]\displaystyle{ 0 \leq x, }[/math] also called the positive cone of the ring) is closed under addition and multiplication, that is, if [math]\displaystyle{ P }[/math] is the set of non-negative elements of a partially ordered ring, then [math]\displaystyle{ P + P \subseteq P }[/math] and [math]\displaystyle{ P \cdot P \subseteq P. }[/math] Furthermore, [math]\displaystyle{ P \cap (-P) = \{0\}. }[/math]

The mapping of the compatible partial order on a ring [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ S \subseteq A }[/math] is a subset of a ring [math]\displaystyle{ A, }[/math] and:

1. [math]\displaystyle{ 0 \in S }[/math]
2. [math]\displaystyle{ S \cap (-S) = \{0\} }[/math]
3. [math]\displaystyle{ S + S \subseteq S }[/math]
4. [math]\displaystyle{ S \cdot S \subseteq S }[/math]

then the relation [math]\displaystyle{ \,\leq\, }[/math] where [math]\displaystyle{ x \leq y }[/math] if and only if [math]\displaystyle{ y - x \in S }[/math] defines a compatible partial order on [math]\displaystyle{ A }[/math] (that is, [math]\displaystyle{ (A, \leq) }[/math] is a partially ordered ring).

In any l-ring, the absolute value [math]\displaystyle{ |x| }[/math] of an element [math]\displaystyle{ x }[/math] can be defined to be [math]\displaystyle{ x \vee(-x), }[/math] where [math]\displaystyle{ x \vee y }[/math] denotes the maximal element. For any [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y, }[/math] [math]\displaystyle{ |x \cdot y| \leq |x| \cdot |y| }[/math] holds.

f-rings

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring [math]\displaystyle{ (A, \leq) }[/math] in which [math]\displaystyle{ x \wedge y = 0 }[/math] and [math]\displaystyle{ 0 \leq z }[/math] imply that [math]\displaystyle{ zx \wedge y = xz \wedge y = 0 }[/math] for all [math]\displaystyle{ x, y, z \in A. }[/math] 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 [math]\displaystyle{ X }[/math] be a Hausdorff space, and [math]\displaystyle{ \mathcal{C}(X) }[/math] be the space of all continuous, real-valued functions on [math]\displaystyle{ X. }[/math] [math]\displaystyle{ \mathcal{C}(X) }[/math] is an Archimedean f-ring with 1 under the following pointwise operations: [math]\displaystyle{ [f + g](x) = f(x) + g(x) }[/math] [math]\displaystyle{ [fg](x) = f(x) \cdot g(x) }[/math] [math]\displaystyle{ [f \wedge g](x) = f(x) \wedge g(x). }[/math]

From an algebraic point of view the rings [math]\displaystyle{ \mathcal{C}(X) }[/math] are fairly rigid. For example, localisations, residue rings or limits of rings of the form [math]\displaystyle{ \mathcal{C}(X) }[/math] 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.

• [math]\displaystyle{ |xy| = |x||y| }[/math] in an f-ring.

• The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.

• 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.

Suppose [math]\displaystyle{ (A, \leq) }[/math] is a commutative ordered ring, and [math]\displaystyle{ x, y, z \in A. }[/math] Then:

	by
The additive group of [math]\displaystyle{ A }[/math] is an ordered group	OrdRing_ZF_1_L4
[math]\displaystyle{ x \leq y \text{ if and only if } x - y \leq 0 }[/math]	OrdRing_ZF_1_L7
[math]\displaystyle{ x \leq y }[/math] and [math]\displaystyle{ 0 \leq z }[/math] imply [math]\displaystyle{ xz \leq yz }[/math] and [math]\displaystyle{ zx \leq zy }[/math]	OrdRing_ZF_1_L9
[math]\displaystyle{ 0 \leq 1 }[/math]	ordring_one_is_nonneg
[math]\displaystyle{ |xy| = |x| |y| }[/math]	OrdRing_ZF_2_L5
[math]\displaystyle{ |x+y| \leq |x| + |y| }[/math]	ord_ring_triangle_ineq
[math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ (A, \leq) }[/math] is closed under multiplication if and only if [math]\displaystyle{ A }[/math] has no zero divisors.	OrdRing_ZF_3_L3
If [math]\displaystyle{ A }[/math] is non-trivial ([math]\displaystyle{ 0 \neq 1 }[/math]), then it is infinite.	ord_ring_infinite

See also

• Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
• Ordered field – Algebraic object with an ordered structure
• Ordered topological vector space
• Ordered vector space – Vector space with a partial order
• Partially ordered space – Partially ordered topological space
• Riesz space – Partially ordered vector space, ordered as a lattice

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

• Hazewinkel, Michiel, ed. (2001), "Ordered ring", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Partially Ordered Ring at PlanetMath.org.