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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:
| | Surely the second option would be more beneficial for any website. The next step is to visit your Word - Press blog dashboard. In case you have any kind of issues regarding where by as well as how you can use [http://xyz.ms/wordpress_backup_697126 wordpress backup], you possibly can e-mail us at our own internet site. These templates are professionally designed and are also Adsense ready. If you are using videos on your site then this is the plugin to use. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. If a newbie missed a certain part of the video then they could always rewind. A Wordpress plugin is a software that you can install into your Wordpress site. You can add new functionalities and edit the existing ones to suit your changing business needs. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. Word - Press has different exciting features including a plug-in architecture with a templating system. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. One of the great features of Wordpress is its ability to integrate SEO into your site. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. Make sure you have the latest versions of all your plugins are updated. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Visit our website to learn more about how you can benefit. Just download it from the website and start using the same. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Customers within a few seconds after visiting a site form their opinion about the site. |
| :<math>x\leq y</math> implies <math>x + z\leq y + z</math>
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| and
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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: | |
| # <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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