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In abstract algebra, a partially ordered group is a group ''(G,+)'' equipped with a partial order "≤" that is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then ''a+g'' ≤ ''b+g'' and ''g+a'' ≤ ''g+b''. An element ''x'' of ''G'' is called positive element if 0 ≤ ''x''. The set of elements 0 ≤ ''x'' is often denoted with ''G''+, and it is called the positive cone of G. So we have ''a'' ≤ ''b'' if and only if ''-a''+''b'' ∈ ''G''+. By the definition, we can reduce the partial order to a monadic property: ''a'' ≤ ''b'' if and only if ''0'' ≤ ''-a''+''b''. For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially ordered group if and only if there exists a subset ''H'' (which is ''G''+) of ''G'' such that: * ''0'' ∈ ''H'' * if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a+b'' ∈ ''H'' * if ''a'' ∈ ''H'' then ''-x''+''a''+''x'' ∈ ''H'' for each ''x'' of ''G'' * if ''a'' ∈ ''H'' and ''-a'' ∈ ''H'' then ''a=0'' A partially ordered group ''G'' with positive cone ''G''+ is said to be unperforated if ''n'' · ''g'' ∈ ''G''+ for some positive integer ''n'' implies ''g'' ∈ ''G''+. Being unperforated means there is no "gap" in the positive cone ''G''+. If the order on the group is a linear order, then it is said to be a linearly ordered group. If the order on the group is a lattice order, i.e. any two elements have a least upper bound, then it is a lattice-ordered group (shortly l-group, though usually typeset with a script ell: ℓ-group). A Riesz group is an unperforated partially ordered group with a property slightly weaker than being a lattice ordered group. Namely, a Riesz group satisfies the Riesz interpolation property: if ''x''1, ''x''2, ''y''1, ''y''2 are elements of ''G'' and ''xi'' ≤ ''yj'', then there exists ''z'' ∈ ''G'' such that ''xi'' ≤ ''z'' ≤ ''yj''. If ''G'' and ''H'' are two partially ordered groups, a map from ''G'' to ''H'' is a ''morphism of partially ordered groups'' if it is both a group homomorphism and a monotonic function. The partially ordered groups, together with this notion of morphism, form a category. Partially ordered groups are used in the definition of valuations of fields. == Examples == * An ordered vector space is a partially ordered group * A Riesz space is a lattice-ordered group * A typical example of a partially ordered group is Z''n'', where the group operation is componentwise addition, and we write (''a''1,...,''a''''n'') ≤ (''b''1,...,''b''''n'') if and only if ''a''''i'' ≤ ''b''''i'' (in the usual order of integers) for all ''i''=1,...,''n''. * More generally, if ''G'' is a partially ordered group and ''X'' is some set, then the set of all functions from ''X'' to ''G'' is again a partially ordered group: all operations are performed componentwise. Furthermore, every subgroup of ''G'' is a partially ordered group: it inherits the order from ''G''. * If ''A'' is an approximately finite-dimensional C *-algebra, or more generally, if ''A'' is a stably finite unital C *-algebra, then K0(''A'') is a partially ordered abelian group. (Elliott, 1976) 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「partially ordered group」の詳細全文を読む スポンサード リンク
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