Partially ordered ring

<h2 id="properties">Properties</h2>
<p>The additive group of a partially ordered ring is always a <a href="/facts/Partially_ordered_group/xoLNpEqH">partially ordered group</a>.
</p><p>The set of non-negative elements of a partially ordered ring (the set of elements 
  
    
      
        x
      
    
    {\displaystyle x}
  
 for which 
  
    
      
        0
        ≤
        x
        ,
      
    
    {\displaystyle 0\leq x,}
  
 also called the positive cone of the ring) is closed under addition and multiplication, that is, if 
  
    
      
        P
      
    
    {\displaystyle P}
  
 is the set of non-negative elements of a partially ordered ring, then 
  
    
      
        P
        +
        P
        ⊆
        P
      
    
    {\displaystyle P+P\subseteq P}
  
 and 
  
    
      
        P
        ⋅
        P
        ⊆
        P
        .
      
    
    {\displaystyle P\cdot P\subseteq P.}
  
 Furthermore, 
  
    
      
        P
        ∩
        (
        −
        P
        )
        =
        {
        0
        }
        .
      
    
    {\displaystyle P\cap (-P)=\{0\}.}

</p><p>The mapping of the compatible partial order on a ring 
  
    
      
        A
      
    
    {\displaystyle A}
  
 to the set of its non-negative elements is <a href="/facts/Injective/5ES6tqf5">one-to-one</a>;<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> 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.
</p><p>If 
  
    
      
        S
        ⊆
        A
      
    
    {\displaystyle S\subseteq A}
  
 is a <a href="/facts/Subset/SJGERmbA">subset</a> of a ring 
  
    
      
        A
        ,
      
    
    {\displaystyle A,}
  
 and:
</p>
<ol><li>
  
    
      
        0
        ∈
        S
      
    
    {\displaystyle 0\in S}
  
</li>
<li>
  
    
      
        S
        ∩
        (
        −
        S
        )
        =
        {
        0
        }
      
    
    {\displaystyle S\cap (-S)=\{0\}}
  
</li>
<li>
  
    
      
        S
        +
        S
        ⊆
        S
      
    
    {\displaystyle S+S\subseteq S}
  
</li>
<li>
  
    
      
        S
        ⋅
        S
        ⊆
        S
      
    
    {\displaystyle S\cdot S\subseteq S}
  
</li></ol>
<p>then the relation 
  
    
      
        
        ≤
        
      
    
    {\displaystyle \,\leq \,}
  
 where 
  
    
      
        x
        ≤
        y
      
    
    {\displaystyle x\leq y}
  
 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> 
  
    
      
        y
        −
        x
        ∈
        S
      
    
    {\displaystyle y-x\in S}
  
 defines a compatible partial order on 
  
    
      
        A
      
    
    {\displaystyle A}
  
 (that is, 
  
    
      
        (
        A
        ,
        ≤
        )
      
    
    {\displaystyle (A,\leq )}
  
 is a partially ordered ring).<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p><p>In any l-ring, the absolute value 
  
    
      
        
          |
        
        x
        
          |
        
      
    
    {\displaystyle |x|}
  
 of an element 
  
    
      
        x
      
    
    {\displaystyle x}
  
 can be defined to be 
  
    
      
        x
        ∨
        (
        −
        x
        )
        ,
      
    
    {\displaystyle x\vee (-x),}
  
 where 
  
    
      
        x
        ∨
        y
      
    
    {\displaystyle x\vee y}
  
 denotes the <a href="/facts/Maximal_element/7FCC9qAW">maximal element</a>. For any 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
        ,
      
    
    {\displaystyle y,}

|
        
        x
        ⋅
        y
        
          |
        
        ≤
        
          |
        
        x
        
          |
        
        ⋅
        
          |
        
        y
        
          |
        
      
    
    {\displaystyle |x\cdot y|\leq |x|\cdot |y|}

holds.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p>
<h2 id="f-rings">f-rings</h2>
<p>An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring 
  
    
      
        (
        A
        ,
        ≤
        )
      
    
    {\displaystyle (A,\leq )}
  
 in which 
  
    
      
        x
        ∧
        y
        =
        0
      
    
    {\displaystyle x\wedge y=0}
  
<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> and 
  
    
      
        0
        ≤
        z
      
    
    {\displaystyle 0\leq z}
  
 imply that 
  
    
      
        z
        x
        ∧
        y
        =
        x
        z
        ∧
        y
        =
        0
      
    
    {\displaystyle zx\wedge y=xz\wedge y=0}
  
 for all 
  
    
      
        x
        ,
        y
        ,
        z
        ∈
        A
        .
      
    
    {\displaystyle x,y,z\in A.}
  
 They were first introduced by <a href="/facts/Garrett_Birkhoff/5jeI7Wmr">Garrett Birkhoff</a> 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.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> The additional hypothesis required of f-rings eliminates this possibility.
</p>
<h3>Example</h3>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a>, and 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 be the <a href="/facts/Topological_space/v2ut7CbK">space</a> of all <a href="/facts/Continuous_function/sKbl02pB">continuous</a>, <a href="/facts/Real_number/R02gw5Pb">real</a>-valued <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> on 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}
  
 is an Archimedean f-ring with 1 under the following pointwise operations:

[
        f
        +
        g
        ]
        (
        x
        )
        =
        f
        (
        x
        )
        +
        g
        (
        x
        )
      
    
    {\displaystyle [f+g](x)=f(x)+g(x)}

[
        f
        g
        ]
        (
        x
        )
        =
        f
        (
        x
        )
        ⋅
        g
        (
        x
        )
      
    
    {\displaystyle [fg](x)=f(x)\cdot g(x)}

[
        f
        ∧
        g
        ]
        (
        x
        )
        =
        f
        (
        x
        )
        ∧
        g
        (
        x
        )
        .
      
    
    {\displaystyle [f\wedge g](x)=f(x)\wedge g(x).}
  
<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p><p>From an algebraic point of view the rings 
  
    
      
        
          
            C
          
        
        (
        X
        )
      
    
    {\displaystyle {\mathcal {C}}(X)}

are fairly rigid. For example, <a href="/facts/Localization_(commutative_algebra)/rjevzBJf">localisations</a>, residue rings or limits of rings of the form 
  
    
      
        
          
            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 <a href="/facts/Real_closed_ring/S12Gep0U">real closed rings</a>.
</p>
<h3>Properties</h3>
<ul><li>A <a href="/facts/Direct_product/GehGQmNV">direct product</a> of f-rings is an f-ring, an l-subring of an f-ring is an f-ring, and an l-homomorphic <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of an f-ring is an f-ring.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li></ul>
<ul><li>
  
    
      
        
          |
        
        x
        y
        
          |
        
        =
        
          |
        
        x
        
          |
        
        
          |
        
        y
        
          |
        
      
    
    {\displaystyle |xy|=|x||y|}
  
 in an f-ring.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li></ul>
<ul><li>The <a href="/facts/Category_(mathematics)/uTvF1odV">category</a> Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li></ul>
<ul><li>Every ordered ring is an f-ring, so every sub-direct union of ordered rings is also an f-ring. Assuming the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>, 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.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> Some mathematicians take this to be the definition of an f-ring.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li></ul>
<h2 id="formally-verified-results-for-commutative-ordered-rings">Formally verified results for commutative ordered rings</h2>
<p>IsarMathLib, a <a href="/facts/Library_(computing)/xRzL3P7q">library</a> for the <a href="/facts/Isabelle_(theorem_prover)/CNbtbX83">Isabelle theorem prover</a>, has formal verifications of a few fundamental results on <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a> ordered rings. The results are proved in the ring1 context.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a>
</p><p>Suppose 
  
    
      
        (
        A
        ,
        ≤
        )
      
    
    {\displaystyle (A,\leq )}
  
 is a commutative ordered ring, and 
  
    
      
        x
        ,
        y
        ,
        z
        ∈
        A
        .
      
    
    {\displaystyle x,y,z\in A.}
  
 Then:
</p>
<table><tbody><tr><th></th><th>by</th></tr><tr><td>The additive group of                     A              {\displaystyle A}   is an ordered group</td><td>OrdRing_ZF_1_L4</td></tr><tr><td>                    x        ≤        y                   if and only if                 x        −        y        ≤        0              {\displaystyle x\leq y{\text{ if and only if }}x-y\leq 0}  </td><td>OrdRing_ZF_1_L7</td></tr><tr><td>                    x        ≤        y              {\displaystyle x\leq y}   and                     0        ≤        z              {\displaystyle 0\leq z}   imply                    x        z        ≤        y        z              {\displaystyle xz\leq yz}   and                     z        x        ≤        z        y              {\displaystyle zx\leq zy}  </td><td>OrdRing_ZF_1_L9</td></tr><tr><td>                    0        ≤        1              {\displaystyle 0\leq 1}  </td><td>ordring_one_is_nonneg</td></tr><tr><td>                              |                x        y                  |                =                  |                x                  |                          |                y                  |                      {\displaystyle |xy|=|x||y|}  </td><td>OrdRing_ZF_2_L5</td></tr><tr><td>                              |                x        +        y                  |                ≤                  |                x                  |                +                  |                y                  |                      {\displaystyle |x+y|\leq |x|+|y|}  </td><td>ord_ring_triangle_ineq</td></tr><tr><td>                    x              {\displaystyle x}   is either in the positive set, equal to 0 or in minus the positive set.</td><td>OrdRing_ZF_3_L2</td></tr><tr><td>The set of positive elements of                     (        A        ,        ≤        )              {\displaystyle (A,\leq )}   is closed under multiplication if and only if                     A              {\displaystyle A}   has no <a href="/facts/Zero_divisor/5GraAU8o">zero divisors</a>.</td><td>OrdRing_ZF_3_L3</td></tr><tr><td>If                     A              {\displaystyle A}   is <a href="/facts/Trivial_ring/trFjtDJ1">non-trivial</a> (                    0        ≠        1              {\displaystyle 0\neq 1}  ), then it is infinite.</td><td>ord_ring_infinite</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Linearly_ordered_group/APXGv4UI">Linearly ordered group</a> – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb</li>
<li><a href="/facts/Ordered_field/an0prSgG">Ordered field</a> – Algebraic object with an ordered structure</li>
<li><a href="/facts/Ordered_group/xoLNpEqH">Ordered group</a> – Group with a compatible partial orderPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Ordered_topological_vector_space/N1fvN8th">Ordered topological vector space</a></li>
<li><a href="/facts/Ordered_vector_space/yN9VfKMa">Ordered vector space</a> – Vector space with a partial order</li>
<li><a href="/facts/Partially_ordered_space/06RwXYXo">Partially ordered space</a> – Partially ordered topological space</li>
<li><a href="/facts/Riesz_space/FX7Mwfxe">Riesz space</a> – Partially ordered vector space, ordered as a lattice</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Birkhoff, G.; R. Pierce (1956). "Lattice-ordered rings". <i>Anais da Academia Brasileira de Ciências</i>. 28: 41–69.</li>
<li>Gillman, Leonard; <a href="/facts/Meyer_Jerison/Sl7yD8T2">Jerison, Meyer</a> Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Ordered_ring">"Ordered ring"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a href="https://planetmath.org/PartiallyOrderedRing">Partially Ordered Ring</a> at <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389. <a href="https://doi.org/10.1007%2FBF02546389" target="_blank">https://doi.org/10.1007%2FBF02546389</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389. <a href="https://doi.org/10.1007%2FBF02546389" target="_blank">https://doi.org/10.1007%2FBF02546389</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389. <a href="https://doi.org/10.1007%2FBF02546389" target="_blank">https://doi.org/10.1007%2FBF02546389</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez (ed.). Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8. <a href="0-7923-4377-8" target="_blank">0-7923-4377-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>∧
      
    
    {\displaystyle \wedge }
  
 denotes infimum. <a href="/wiki/Infimum" target="_blank">/wiki/Infimum</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389. <a href="https://doi.org/10.1007%2FBF02546389" target="_blank">https://doi.org/10.1007%2FBF02546389</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389. <a href="https://doi.org/10.1007%2FBF02546389" target="_blank">https://doi.org/10.1007%2FBF02546389</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez (ed.). Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8. <a href="0-7923-4377-8" target="_blank">0-7923-4377-8</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez (ed.). Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8. <a href="0-7923-4377-8" target="_blank">0-7923-4377-8</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Hager, Anthony W.; Jorge Martinez (2002). "Functorial rings of quotients—III: The maximum in Archimedean f-rings". Journal of Pure and Applied Algebra. 169: 51–69. doi:10.1016/S0022-4049(01)00060-3. <a href="https://doi.org/10.1016%2FS0022-4049%2801%2900060-3" target="_blank">https://doi.org/10.1016%2FS0022-4049%2801%2900060-3</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings". Acta Mathematica. 104 (3–4): 163–215. doi:10.1007/BF02546389. <a href="https://doi.org/10.1007%2FBF02546389" target="_blank">https://doi.org/10.1007%2FBF02546389</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez (ed.). Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8. <a href="0-7923-4377-8" target="_blank">0-7923-4377-8</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
<li id="fn:16"><p>"IsarMathLib" (PDF). Retrieved 2009-03-31. <a href="http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf" target="_blank">http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li>
</ol>

Partially ordered ring open-in-new

Partially ordered ring