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Concept
Linearly ordered group
Summary
In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a 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 ≤ b implies ca ≤ cb for all a, b, c in G,
right-ordered group if ≤ is right-invariant, that is a ≤ b implies ac ≤ bc for all a, b, 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.
In this section is a left-invariant order on a group with identity element . All that is said applies to right-invariant orders with the obvious modifications. Note that being left-invariant is equivalent to the order defined by if and only if being right-invariant. In particular a group being left-orderable is the same as it being right-orderable.
In analogy with ordinary numbers we call an element of an ordered group positive if . The set of positive elements in an ordered group is called the positive cone, it is often denoted with ; the slightly different notation is used for the positive cone together with the identity element.
The positive cone characterises the order ; indeed, by left-invariance we see that if and only if . In fact a left-ordered group can be defined as a group together with a subset satisfying the two conditions that:
for we have also ;
let , then is the disjoint union of and .
The order associated with is defined by ; the first condition amounts to left-invariance and the second to the order being well-defined and total. The positive cone of is .
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