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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 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''+.〔Note that the + is written as a subscript, to distinguish from ''G''+ which includes the identity element. See e.g. (IsarMathLib ), p. 344.〕 For every element ''a'' of a linearly ordered group ''G'' either ''a'' ∈ ''G''+, or '' -a'' ∈ ''G''+, or ''a'' = 0. If a linearly ordered group ''G'' is not trivial (i.e. 0 is not its only element), then ''G''+ is infinite. 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: : If in addition the group ''G'' is abelian, then for any ''a'', ''b'' ∈ ''G'' the triangle inequality is satisfied: |''a'' + ''b''| ≤ |''a''| + |''b''|. ==Examples== Any totally ordered group is torsion-free. Conversely, F. W. Levi showed that an abelian group admits a linear order if and only if it is torsion free . 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, . 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 . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「linearly ordered group」の詳細全文を読む スポンサード リンク
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