partially ordered ring

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 $0\; \backslash leq\; x,$ 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 $P\; +\; P\; \backslash subseteq\; P$ and $P\; \backslash cdot\; P\; \backslash subseteq\; P.$ Furthermore, $P\; \backslash cap\; (-P)\; =\; \backslash \{0\backslash \}.$

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 $S\; \backslash subseteq\; A$ is a subset of a ring $A,$ and:

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

then the relation $\backslash ,\backslash leq\backslash ,$ where $x\; \backslash leq\; y$ if and only if $y\; -\; x\; \backslash in\; S$ defines a compatible partial order on $A$ (that is, $(A,\; \backslash leq)$ is a partially ordered ring).

In any l-ring, the {{em|absolute value}} $|x|$ of an element $x$ can be defined to be $x\; \backslash vee(-x),$ where $x\; \backslash vee\; y$ denotes the maximal element. For any $x$ and $y,$

$$|x\; \backslash cdot\; y|\; \backslash leq\; |x|\; \backslash cdot\; |y|$$

holds.{{cite book| last = Henriksen | first = Melvin | authorlink = Melvin Henriksen | chapter = A survey of f-rings and some of their generalizations | pages = 1–26 | title = Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995 | year = 1997 | editor = W. Charles Holland and Jorge Martinez | isbn = 0-7923-4377-8 | publisher = Kluwer Academic Publishers | location = the Netherlands}}

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring $(A,\; \backslash leq)$ in which $x\; \backslash wedge\; y\; =\; 0$$\backslash wedge$ denotes infimum. and $0\; \backslash leq\; z$ imply that $zx\; \backslash wedge\; y\; =\; xz\; \backslash wedge\; y\; =\; 0$ for all $x,\; y,\; z\; \backslash in\; A.$ 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.

Let $X$ be a Hausdorff space, and $\backslash mathcal\{C\}(X)$ be the space of all continuous, real-valued functions on $X.$ $\backslash mathcal\{C\}(X)$ 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 $\backslash mathcal\{C\}(X)$

are fairly rigid. For example, localisations, residue rings or limits of rings of the form $\backslash mathcal\{C\}(X)$ 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.

• $|xy|\; =\; |x||y|$ in an f-ring.

• The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.{{cite journal| last = Hager | first = Anthony W. |author2=Jorge Martinez | year = 2002 | title = Functorial rings of quotients—III: The maximum in Archimedean f-rings | journal = Journal of Pure and Applied Algebra | volume = 169 | pages = 51–69| doi = 10.1016/S0022-4049(01)00060-3| doi-access = free }}

• 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.

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.{{cite web| url = http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf | title = IsarMathLib | accessdate = 2009-03-31}}

Suppose $(A,\; \backslash leq)$ is a commutative ordered ring, and $x,\; y,\; z\; \backslash in\; A.$ Then:

• {{annotated link|Linearly ordered group}}
• {{annotated link|Ordered field}}
• {{annotated link|Ordered group}}
• {{annotated link|Ordered topological vector space}}
• {{annotated link|Ordered vector space}}
• {{annotated link|Partially ordered space}}
• {{annotated link|Riesz space}}

{{reflist}}

• {{cite journal| last = Birkhoff | first = G. |author2=R. Pierce | year = 1956 | title = Lattice-ordered rings | journal = Anais da Academia Brasileira de Ciências | volume = 28 | pages = 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

• {{SpringerEOM| title = Ordered ring}}
• {{PlanetMath| title = Partially Ordered Ring | urlname = PartiallyOrderedRing}}

Category:Ring theory

Category:Ordered algebraic structures