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Note that sometimes the term ''ordered group'' is used for a linearly (or totally) ordered group, and what we describe here is called a ''partially ordered group''. By the definition we can reduce the partial order to a monadic property: ''a'' ≤ ''b'' iff ''1'' ≤ ''a''-1 ''b''. The set of elements ''x'' ≥ ''1'' of ''G'' are often denoted with ''G''+. So we have ''a'' ≤ ''b'' there exists a subset ''H'' (which is ''G''+) of ''G'' such that:
A typical example of an ordered group is Z ''n'', where the group operation is componentwise addition, and we write (''a''1,...,''a''''n'') ≤ (''b''1,...,''b''''n'') Iff ''a''''i'' ≤ ''b''''i'' (in the usual order of integers) for all ''i''=1,...,''n''. More generally, if ''G'' is an ordered group and ''X'' is some set, then the set of all functions from ''X'' to ''G'' is again an ordered group: all operations are performed componentwise. Furthermore, every Subgroup of ''G'' is an ordered group: it inherits the order from ''G''. If the order on the group is a Linear Order , we speak of a Linearly Ordered Group . If ''G'' and ''H'' are two ordered groups, a map from ''G'' to ''H'' is a ''morphism of ordered groups'' if it is both a Group Homomorphism and a Monotonic Function . The ordered groups, together with this notion of morphism, form a Category . Ordered groups are used in the definition of Valuation s of Field s. |
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