partially ordered ring
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abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a partially ordered ring is a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
(''A'', +, ·), together with a ''compatible partial order'', that is, a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
\,\leq\, on the underlying set ''A'' that is compatible with the ring operations in the sense that it satisfies: x \leq y \text x + z \leq y + z and 0 \leq x \text 0 \leq y \text 0 \leq x \cdot y for all x, y, z\in A. 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 (A, \leq) where partially ordered additive
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
is Archimedean. An ordered ring, also called a totally ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflex ...
. An l-ring, or lattice-ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a
lattice order A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper boun ...
.


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 x for which 0 \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 \subseteq P and P \cdot P \subseteq P. Furthermore, P \cap (-P) = \. 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 \subseteq A is a subset of a ring A, and: # 0 \in S # S \cap (-S) = \ # S + S \subseteq S # S \cdot S \subseteq S then the relation \,\leq\, where x \leq y
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
y - x \in S defines a compatible partial order on A (that is, (A, \leq) is a partially ordered ring). In any l-ring, the , x, of an element x can be defined to be x \vee(-x), where x \vee y denotes the maximal element. For any x and y, , x \cdot y, \leq , x, \cdot , y, holds.


f-rings

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring (A, \leq) in which x \wedge y = 0 and 0 \leq z imply that zx \wedge y = xz \wedge y = 0 for all x, y, z \in A. They were first introduced by
Garrett Birkhoff Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician Ge ...
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 X be a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
, and \mathcal(X) be the
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
of all continuous,
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s on X. \mathcal(X) is an Archimedean f-ring with 1 under the following pointwise operations: + gx) = f(x) + g(x) gx) = f(x) \cdot g(x) \wedge gx) = f(x) \wedge g(x). From an algebraic point of view the rings \mathcal(X) are fairly rigid. For example, localisations, residue rings or limits of rings of the form \mathcal(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. * , xy, = , x, , y, in an f-ring. * The
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, 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 A library is a collection of materials, books or media that are accessible for use and not just for display purposes. A library provides physical (hard copies) or digital access (soft copies) materials, and may be a physical location or a vir ...
for the
Isabelle theorem prover The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As an LCF-style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs withou ...
, has formal verifications of a few fundamental results on
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
ordered rings. The results are proved in the ring1 context. Suppose (A, \leq) is a commutative ordered ring, and x, y, z \in A. Then:


See also

* * * * * * *


References


Further reading

* * 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

* * {{PlanetMath, title = Partially Ordered Ring , urlname = PartiallyOrderedRing Ring theory Ordered algebraic structures