Linearly ordered group
In 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. Let a, b, c ∈ G, we say that (G, ≤) is a
- left-ordered group if a ≤ b implies c+a ≤ c+b
- right-ordered group if a ≤ b implies a+c ≤ b+c
- bi-ordered group if it is both left-ordered and right-ordered
Note that G need not be abelian, even though we use additive notation (+) for the group operation.
In analogy with ordinary numbers, we call an element c of an ordered group positive if 0 ≤ c and c ≠ 0, where "0" here denotes the identity element of the group (not necessarily the familiar zero of the real numbers). The set of positive elements in a group is often denoted with G+.[a]
Elements of a linearly ordered group satisfy trichotomy: every element a of a linearly ordered group G is either positive (a ∈ G+), negative (−a ∈ G+), or zero (a = 0). If a linearly ordered group G is not trivial (i.e. 0 is not its only element), then G+ is infinite, since all multiples of a non-zero element are distinct.[b] Therefore, every nontrivial linearly ordered group is infinite.
If a is an element of a linearly ordered group G, then the absolute value of a, denoted by |a|, is defined to be:
Otto Hölder showed that every Archimedean group (a bi-ordered group satisfying an Archimedean property) is isomorphic to a subgroup of the additive group of real numbers, (Fuchs & Salce 2001, p. 61). If we write the Archimedean l.o. group multiplicatively, this may be shown by considering the Dedekind completion, of the closure of an l.o. group under th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each the exponential maps are well defined order preserving/reversing, topological group isomorphisms. Completing an l.o. group can be difficult in the non-Archimedean case. In these cases, one may classify a group by its rank: which is related to the order type of the largest sequence of convex subgroups.
A large source of examples of left-orderable groups comes from groups acting on the real line by order preserving homeomorphisms. Actually, for countable groups, this is known to be a characterization of left-orderability, see for instance (Ghys 2001).
- Note that the + is written as a subscript, to distinguish from G+ which includes the identity element. See e.g. IsarMathLib, p. 344.
- Formally, given any non-zero element c (which we can assume to be positive, otherwise take −c) and natural number k we have , so by induction, given two natural numbers k < l, we have , so there is an injection from the natural numbers into G.
- Levi, F.W. (1942), "Ordered groups.", Proc. Indian Acad. Sci., A16: 256–263
- Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, 84, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1963-0, MR 1794715
- Ghys, É. (2001), "Groups acting on the circle.", L´Eins. Math., 47: 329–407