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noowutah
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I have a total order on a multi-dimensional, real-numbered vector space X. This means that for any vectors x,y\in{}X either xRy or yRx. Total orders are usually transitive, total, and antisymmetric (if xRy and yRx then x=y), but this one is not necessarily antisymmetric, it is only transitive and total. Which further conditions do I need to prove that it is antisymmetric, ie. xRy and yRx imply x=y just based on the fact that the ordering is total and transitive? One example of a transitive, total ordering which is not antisymmetric is one that uses only one coordinate of the vector and the usual less than (\geq) or greater than (\leq) relations: xRy if and only if x_{i}>y_{i} for some fixed i, and we ignore all other coordinates, all x_{j} for j\neq{}i. I have a hunch that ALL transitive, total orderings that are not antisymmetric are of this type. How could I formalize and prove this claim?
The claim, in other words, is that the only transitive, total, not-antisymmetric orderings R of a n-dimensional, real-numbered vector space are those which only consider one dimension and disregard the others. Let R be a such an ordering. Then there exists a transitive, total, antisymmetric ordering R' of the real numbers such that for a fixed k\in\{1,\ldots,n\} xRy if and only if x_{k}R'y_{k}. How would I prove this (and is it correct)?
The claim, in other words, is that the only transitive, total, not-antisymmetric orderings R of a n-dimensional, real-numbered vector space are those which only consider one dimension and disregard the others. Let R be a such an ordering. Then there exists a transitive, total, antisymmetric ordering R' of the real numbers such that for a fixed k\in\{1,\ldots,n\} xRy if and only if x_{k}R'y_{k}. How would I prove this (and is it correct)?
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