Editing Linearly ordered group (section)

{{Short description|Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb}}
In [[mathematics]], specifically [[abstract algebra]], a '''linearly ordered''' or '''totally ordered group''' is a [[group (mathematics)|group]] ''G'' equipped with a [[total order]] "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a:

* '''left-ordered group''' if ≤ is left-invariant, that is ''a''&nbsp;≤&nbsp;''b'' implies ''ca''&nbsp;≤&nbsp;''cb'' for all ''a'',&nbsp;''b'',&nbsp;''c'' in ''G'',
* '''right-ordered group''' if ≤ is right-invariant, that is  ''a''&nbsp;≤&nbsp;''b'' implies ''ac''&nbsp;≤&nbsp;''bc'' for all ''a'',&nbsp;''b'',&nbsp;''c'' in ''G'',
* '''bi-ordered group''' if ≤ is bi-invariant, that is it is both left- and right-invariant. 

A group ''G'' is said to be '''left-orderable''' (or '''right-orderable''', or '''bi-orderable''') if there exists a left- (or right-, or bi-) invariant order on ''G''. 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.