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- This revision was by author genewardsmith and made on 2011-06-25 15:39:21 UTC.
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Original Wikitext content:
[[toc|flat]] =Definition= Given a reduced list of [[Prime limit|p-limit]] vals V, we may define a set of //transversal generators// for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By //reduced// is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r. If v1, v2, ... vr are the vals of V and t1, t2, ... tr are the corresponding transversal generators, then for any p-limit q we have q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q) In this way the transversal generators provide a [[Transverals|transversal]] of the p-limit, and hence the name. =Examples=
Original HTML content:
<html><head><title>Transversal generators</title></head><body><!-- ws:start:WikiTextTocRule:4:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:4 --><!-- ws:start:WikiTextTocRule:5: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:5 --><!-- ws:start:WikiTextTocRule:6: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --> <!-- ws:end:WikiTextTocRule:7 --><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> Given a reduced list of <a class="wiki_link" href="/Prime%20limit">p-limit</a> vals V, we may define a set of <em>transversal generators</em> for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By <em>reduced</em> is meant that the gcd of the elements of each of the vals is 1--or in other words, none of the vals are contorted--and that they are linearly independent, so that if there are r vals, the rank of V as a matrix is r.<br /> <br /> If v1, v2, ... vr are the vals of V and t1, t2, ... tr are the corresponding transversal generators, then for any p-limit q we have<br /> <br /> q = t1^v1(q) * t2^v2(q) * ... * tr^vr(q)<br /> <br /> In this way the transversal generators provide a <a class="wiki_link" href="/Transverals">transversal</a> of the p-limit, and hence the name.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> </body></html>