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| In [[abstract algebra]], a '''partially ordered ring''' is a [[Ring (mathematics)|ring]] (''A'', +, '''·''' ), together with a ''compatible partial order'', i.e. a [[partial order]] <math>\leq</math> on the underlying set ''A'' that is compatible with the ring operations in the sense that it satisfies:
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| :<math>x\leq y</math> implies <math>x + z\leq y + z</math>
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| :<math>0\leq x</math> and <math>0\leq y</math> imply that <math>0\leq x\cdot y</math>
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| for all <math>x, y, z\in A</math>.<ref name="Anderson">{{cite journal| last = Anderson | first = F. W. | title = Lattice-ordered rings of quotients | journal = Canadian Journal of Mathematics | pages = 434–448}}</ref> 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 <math>(A, \leq)</math> where <math>A</math>'s partially ordered additive [[Partially ordered group|group]] is [[Archimedean group|Archimedean]].<ref name="Johnson">{{cite journal| last = Johnson | first = D. G. | year = 1960 | month = December | title = A structure theory for a class of lattice-ordered rings | journal = Acta Mathematica | volume = 104 | issue = 3–4 | pages = 163–215 | doi = 10.1007/BF02546389}}</ref> | |
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| An '''ordered ring''', also called a '''totally ordered ring''', is a partially ordered ring <math>(A, \leq)</math> where <math>\le</math> is additionally a [[total order]].<ref name="Anderson" /><ref name="Johnson" />
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| An '''l-ring''', or '''lattice-ordered ring''', is a partially ordered ring <math>(A, \leq)</math> where <math>\leq</math> is additionally a [[lattice order]].
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| == Properties ==
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| The additive group of a partially ordered ring is always a [[partially ordered group]].
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| The set of non-negative elements of a partially ordered ring (the set of elements ''x'' for which <math>0\leq x</math>, also called the positive cone of the ring) is closed under addition and multiplication, i.e., if ''P'' is the set of non-negative elements of a partially ordered ring, then <math>P + P \subseteq P</math>, and <math>P\cdot P \subseteq P</math>. Furthermore, <math>P\cap(-P) = \{0\}</math>.
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| The mapping of the compatible partial order on a ring ''A'' to the set of its non-negative elements is [[bijection|one-to-one]];<ref name="Anderson" /> 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.
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| If ''S'' is a subset of a ring ''A'', and:
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| # <math>0\in S</math>
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| # <math>S\cap(-S) = \{0\}</math>
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| # <math>S + S\subseteq S</math>
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| # <math>S\cdot S\subseteq S</math>
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| then the relation <math>\leq</math> where <math>x\leq y</math> [[iff]] <math>y - x\in S</math> defines a compatible partial order on ''A'' (''ie.'' <math>(A, \leq)</math> is a partially ordered ring).<ref name="Johnson" />
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| In any l-ring, the ''absolute value'' <math>|x|</math> of an element ''x'' can be defined to be <math>x\vee(-x)</math>, where <math>x\vee y</math> denotes the [[maximal element]]. For any ''x'' and ''y'',
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| :<math>|x\cdot y|\leq|x|\cdot|y|</math>
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| holds.<ref name="Henriksen">{{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}}</ref>
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| == f-rings ==
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| An '''f-ring''', or '''Pierce–Birkhoff ring''', is a lattice-ordered ring <math>(A, \leq)</math> in which <math>x\wedge y = 0</math><ref><math>\wedge</math> denotes [[infimum]].</ref> and <math>0\leq z</math> imply that <math>zx\wedge y = xz\wedge y = 0</math> for all <math>x, y, z\in A</math>. 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 negative, even though being a square.<ref name="Johnson" /> The additional hypothesis required of f-rings eliminates this possibility.
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| === Example ===
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| Let ''X'' be a [[Hausdorff space]], and <math>\mathcal{C}(X)</math> be the space of all [[Continuous function|continuous]], [[Real number|real]]-valued [[Function (mathematics)|function]]s on ''X''. <math>\mathcal{C}(X)</math> is an Archimedean f-ring with 1 under the following point-wise operations:
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| :<math>[f + g](x) = f(x) + g(x)</math>
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| :<math>[fg](x) = f(x)\cdot g(x)</math>
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| :<math>[f\wedge g](x) = f(x)\wedge g(x).</math><ref name="Johnson" />
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| From an algebraic point of view the rings <math>\mathcal{C}(X)</math>
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| are fairly rigid. For example localisations, residue rings or limits of
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| rings of the form <math>\mathcal{C}(X)</math> are not of this form in general.
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| A much more flexible class of f-rings containing all rings of continuous functions
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| and resembling many of the properties of these rings, is the class of [[real closed ring]]s.
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| === Properties ===
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| 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 (mathematics)|image]] of an f-ring is an f-ring.<ref name="Henriksen" />
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| <math>|xy|=|x||y|</math> in an f-ring.<ref name="Henriksen" /> | |
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| The [[Category (mathematics)|category]] '''Arf''' consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.<ref name="Hager">{{cite journal| last = Hager | first = Anthony W. | coauthors = 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}}</ref>
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| Every ordered ring is an f-ring, so every subdirect 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 subdirect union of ordered rings.<ref name="Johnson" /> Some mathematicians take this to be the definition of an f-ring.<ref name="Henriksen" />
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| == Formally verified results for commutative ordered rings ==
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| [[IsarMathLib]], a [[Library (computing)|library]] for the [[Isabelle (theorem prover)|Isabelle theorem prover]], has formal verifications of a few fundamental results on [[Commutative ring|commutative]] ordered rings. The results are proved in the <tt>ring1</tt> context.<ref>{{cite web| url = http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf | title = IsarMathLib | accessdate = 2009-03-31}}</ref>
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| Suppose <math>(A, \leq)</math> is a commutative ordered ring, and <math>x, y, z\in A</math>. Then:
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| {| class="wikitable"
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| !
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| ! by
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| |-
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| | The additive group of ''A'' is an ordered group
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| | <tt>OrdRing_ZF_1_L4</tt>
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| |-
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| | <math>x\leq y</math> iff <math>x - y\leq 0</math>
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| | <tt>OrdRing_ZF_1_L7</tt>
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| |-
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| | <math>x\leq y</math> and <math>0\leq z</math> imply<br/><math>xz\leq yz</math> and <math>zx\leq zy</math>
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| | <tt>OrdRing_ZF_1_L9</tt>
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| |-
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| | <math>0\leq 1</math>
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| | <tt>ordring_one_is_nonneg</tt>
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| |-
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| | <math>|xy|=|x||y|</math>
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| | <tt>OrdRing_ZF_2_L5</tt>
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| |-
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| | <math>|x+y|\leq|x|+|y|</math>
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| | <tt>ord_ring_triangle_ineq</tt>
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| |-
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| | ''x'' is either in the positive set, equal to 0, or in minus the positive set.
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| | <tt>OrdRing_ZF_3_L2</tt>
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| |-
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| | The set of positive elements of <math>(A, \leq)</math> is closed under multiplication iff ''A'' has no [[zero divisor]]s.
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| | <tt>OrdRing_ZF_3_L3</tt>
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| |-
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| | If ''A'' is non-trivial (<math>0\neq 1</math>), then it is infinite.
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| | <tt>ord_ring_infinite</tt>
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| |}
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| == References ==
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| <references/>
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| == Further reading ==
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| * {{cite journal| last = Birkhoff | first = G. | coauthors = R. Pierce | year = 1956 | title = Lattice-ordered rings | journal = Anais da Academia Brasileira de Ciências | volume = 28 | pages = 41–69}}
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| * Gillman, Leonard; [[Meyer Jerison|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
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| == External links ==
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| * {{cite web| title = Ordered Ring, Partially Ordered Ring | publisher = [[Encyclopedia of Mathematics]] | url = http://eom.springer.de/O/o070140.htm | accessdate = 2009-04-03}}
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| * {{cite web| title = Partially Ordered Ring | publisher = [[PlanetMath]] | url = http://planetmath.org/encyclopedia/PartiallyOrderedRing.html | accessdate = 2009-03-30}}
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| [[Category:Ring theory]]
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| [[Category:Ordered algebraic structures]]
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