Partially ordered ring

In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order ${\displaystyle \,\leq \,}$ on the underlying set A that is compatible with the ring operations in the sense that it satisfies: $${\displaystyle x\leq y{\text{ implies }}x+z\leq y+z}$$ and $${\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y}$$ 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 ${\displaystyle x}$ for which ${\displaystyle 0\leq x,}$ also called the positive cone of the ring) is closed under addition and multiplication, that is, if ${\displaystyle P}$ 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 ${\displaystyle A}$ 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 ${\displaystyle S\subseteq A}$ is a subset of a ring ${\displaystyle A,}$ 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 ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in S}$ defines a compatible partial order on ${\displaystyle A}$ (that is, ${\displaystyle (A,\leq )}$ is a partially ordered ring).^[2]

In any l-ring, the absolute value ${\displaystyle |x|}$ of an element ${\displaystyle x}$ can be defined to be ${\displaystyle x\vee (-x),}$ where ${\displaystyle x\vee y}$ denotes the maximal element. For any ${\displaystyle x}$ and ${\displaystyle y,}$ $${\displaystyle |x\cdot y|\leq |x|\cdot |y|}$$ 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 not positive, even though it is a square.^[2] The additional hypothesis required of f-rings eliminates this possibility.

Example

Let ${\displaystyle X}$ be a Hausdorff space, and ${\displaystyle {\mathcal {C}}(X)}$ be the space of all continuous, real-valued functions on ${\displaystyle X.}$ ${\displaystyle {\mathcal {C}}(X)}$ 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 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.^[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 ${\displaystyle A}$ is an ordered group	OrdRing_ZF_1_L4
${\displaystyle x\leq y{\text{ if and only if }}x-y\leq 0}$	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
${\displaystyle x}$ 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 if and only if ${\displaystyle A}$ has no zero divisors.	OrdRing_ZF_3_L3
If ${\displaystyle A}$ is non-trivial (${\displaystyle 0\neq 1}$), 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 group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• 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