Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Linearly ordered group
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a linearly ordered or totally ordered group is a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> <i>G</i> equipped with a <a href="/facts/Total_order/xnTtmuYJ">total order</a> "≤" that is <i>translation-invariant</i>. This may have different meanings. We say that (<i>G</i>, ≤) is a: </p> <ul><li>left-ordered group if ≤ is left-invariant, that is <i>a</i> ≤ <i>b</i> implies <i>ca</i> ≤ <i>cb</i> for all <i>a</i>, <i>b</i>, <i>c</i> in <i>G</i>,</li> <li>right-ordered group if ≤ is right-invariant, that is <i>a</i> ≤ <i>b</i> implies <i>ac</i> ≤ <i>bc</i> for all <i>a</i>, <i>b</i>, <i>c</i> in <i>G</i>,</li> <li>bi-ordered group if ≤ is bi-invariant, that is it is both left- and right-invariant.</li></ul> <p>A group <i>G</i> is said to be left-orderable (or right-orderable, or bi-orderable) if there exists a left- (or right-, or bi-) invariant order on <i>G</i>. A simple necessary condition for a group to be left-orderable is to have no elements of finite order; however this is not a sufficient condition. It is equivalent for a group to be left- or right-orderable; however there exist left-orderable groups which are not bi-orderable. </p>