Difference between revisions of "Vector ring"
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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 $a_\lambda \ge 0$. | + | {{TEX|done}}{{MSC|06F25}} |
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| + | A partially ordered ring $R$ (cf. [[Partially ordered set]], [[Ordered ring]]) 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 $R_\lambda$. | 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 $R_\lambda$. | ||
Latest revision as of 21:01, 22 December 2014
2020 Mathematics Subject Classification: Primary: 06F25 [MSN][ZBL]
A partially ordered ring $R$ (cf. Partially ordered set, Ordered ring) 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
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