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 ${\displaystyle ,円\leq ,円}${\displaystyle ,円\leq ,円} on the underlying set A that is compatible with the ring operations in the sense that it satisfies: $${\displaystyle x\leq y{\text{ implies }}x+z\leq y+z}$${\displaystyle x\leq y{\text{ implies }}x+z\leq y+z} and $${\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y}$${\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y} for all ${\displaystyle x,y,z\in A}${\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 ${\displaystyle (A,\leq )}${\displaystyle (A,\leq )} where ${\displaystyle A}${\displaystyle A}'s partially ordered additive group is Archimedean.^[2]

An ordered ring, also called a totally ordered ring, is a partially ordered ring ${\displaystyle (A,\leq )}${\displaystyle (A,\leq )} where ${\displaystyle ,円\leq ,円}${\displaystyle ,円\leq ,円} is additionally a total order.^{[1] [2]}

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

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

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

then the relation ${\displaystyle ,円\leq ,円}${\displaystyle ,円\leq ,円} where ${\displaystyle x\leq y}${\displaystyle x\leq y} if and only if ${\displaystyle y-x\in S}${\displaystyle y-x\in S} defines a compatible partial order on ${\displaystyle A}${\displaystyle A} (that is, ${\displaystyle (A,\leq )}${\displaystyle (A,\leq )} is a partially ordered ring).^[2]

In any l-ring, the absolute value ${\displaystyle |x|}${\displaystyle |x|} of an element ${\displaystyle x}${\displaystyle x} can be defined to be ${\displaystyle x\vee (-x),}${\displaystyle x\vee (-x),} where ${\displaystyle x\vee y}${\displaystyle x\vee y} denotes the maximal element. For any ${\displaystyle x}${\displaystyle x} and ${\displaystyle y,}${\displaystyle y,} $${\displaystyle |x\cdot y|\leq |x|\cdot |y|}$${\displaystyle |x\cdot y|\leq |x|\cdot |y|} holds.^[3]

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring ${\displaystyle (A,\leq )}${\displaystyle (A,\leq )} in which ${\displaystyle x\wedge y=0}${\displaystyle x\wedge y=0}^[4] and ${\displaystyle 0\leq z}${\displaystyle 0\leq z} imply that ${\displaystyle zx\wedge y=xz\wedge y=0}${\displaystyle zx\wedge y=xz\wedge y=0} for all ${\displaystyle x,y,z\in A.}${\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.

Let ${\displaystyle X}${\displaystyle X} be a Hausdorff space, and ${\displaystyle {\mathcal {C}}(X)}${\displaystyle {\mathcal {C}}(X)} be the space of all continuous, real-valued functions on ${\displaystyle X.}${\displaystyle X.} ${\displaystyle {\mathcal {C}}(X)}${\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 [f+g](x)=f(x)+g(x)} $${\displaystyle [fg](x)=f(x)\cdot g(x)}$${\displaystyle [fg](x)=f(x)\cdot g(x)} $${\displaystyle [f\wedge g](x)=f(x)\wedge g(x).}$${\displaystyle [f\wedge g](x)=f(x)\wedge g(x).}^[2]

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

• ${\displaystyle |xy|=|x||y|}${\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]

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 ${\displaystyle (A,\leq )}${\displaystyle (A,\leq )} is a commutative ordered ring, and ${\displaystyle x,y,z\in A.}${\displaystyle x,y,z\in A.} Then:

• Linearly ordered group – Group with translationally invariant total order
• 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

• 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

• "Ordered ring", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Partially Ordered Ring at PlanetMath.

• Ring theory
• Ordered algebraic structures

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• Pages displaying short descriptions of redirect targets via Module:Annotated link