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Difference between revisions of "Vector ring"

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A partially ordered ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965101.png" /> (cf. [[Partially ordered set|Partially ordered set]]) which is a subdirect sum of totally ordered rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965102.png" /> (cf. [[Totally ordered set|Totally ordered set]]). Each element of a vector ring is a vector
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A partially ordered ring $R$ (cf. [[Partially ordered set]]) which is expressible as a subdirect sum of totally ordered rings $R_\lambda$ (cf. [[Totally ordered set]]). Each element of a vector ring is a vector $\mathbf{a} = (\ldots,a_\lambda,\ldots)$with coordinates in $R_\lambda$, and $\mathbf{a} \ge 0$ if and only if each $a_\lambda \ge 0$.
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965103.png" /></td> </tr></table>
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If the partial order of $R$ is an intersection of total orders, then $R$ is a vector ring, and $R$ itself, provided with various linear extensions of its partial order, may be taken as $R_\lambda$.
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with coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965105.png" /> if and only if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965106.png" />. If the partial order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965107.png" /> is an intersection of total orders, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965108.png" /> will be a vector ring, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v0965109.png" /> itself, provided with various linear extensions of its partial order, may be taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096510/v09651010.png" />.
 
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR>
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</table>

Revision as of 20:30, 22 December 2014

A partially ordered ring $R$ (cf. Partially ordered set) which is expressible as a subdirect sum of totally ordered rings $R_\lambda$ (cf. Totally ordered set). Each element of a vector ring is a vector $\mathbf{a} = (\ldots,a_\lambda,\ldots)$with coordinates in $R_\lambda$, and $\mathbf{a} \ge 0$ if and only if each $a_\lambda \ge 0$.

If the partial order of $R$ is an intersection of total orders, then $R$ is a vector ring, and $R$ itself, provided with various linear extensions of its partial order, may be taken as $R_\lambda$.

References

[1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
How to Cite This Entry:
Vector ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_ring&oldid=35820
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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