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.¹²

An l-ring, or lattice-ordered ring, is a partially ordered ring ${\displaystyle (A,\leq )}$ where ${\displaystyle \,\leq \,}$ is additionally a lattice order.

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

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",⁵ 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.

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.

• 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.⁶

• 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