Partially ordered group: Difference between revisions
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In [[abstract algebra]], an '''ordered group''' is a [[group (mathematics)|group]] ''G'' equipped with a [[partial order]] "≤" which is ''translation-invariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then '' |
In [[abstract algebra]], an '''ordered group''' is a [[group (mathematics)|group]] ''(G,+)'' equipped with a [[partial order]] "≤" which 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''. 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''. |
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By the definition, we can reduce the partial order to a monadic property: ''a'' ≤ ''b'' if and only if '' |
By the definition, we can reduce the partial order to a monadic property: ''a'' ≤ ''b'' if and only if ''0'' ≤ ''a''<sup>-1</sup> ''b''. The set of elements ''x'' ≥ ''0'' of ''G'' is often denoted with ''G''<sup>+</sup>, and it is called the ''positive cone of G''. So we have ''a'' ≤ ''b'' [[if and only if]] ''-a''+''b'' ∈ ''G''<sup>+</sup>. |
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The order of an ordered group ''G'' is defined by ''G''<sup>+</sup>; a group is an ordered group [[if and only if]] there exists a subset ''H'' (which is ''G''<sup>+</sup>) of ''G'' such that: |
The order of an ordered group ''G'' is defined by ''G''<sup>+</sup>; a group is an ordered group [[if and only if]] there exists a subset ''H'' (which is ''G''<sup>+</sup>) of ''G'' such that: |
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* '' |
* ''0'' ∈ ''H'' |
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* if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then '' |
* if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a+b'' ∈ ''H'' |
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* if ''a'' ∈ ''H'' then ''x'' |
* if ''a'' ∈ ''H'' then ''-x''+''a''+''x'' ∈ ''H'' for each ''x'' of ''G'' |
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* if ''a'' ∈ ''H'' and ''a'' |
* if ''a'' ∈ ''H'' and ''-a'' ∈ ''H'' then ''a=0'' |
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If the order on the group is a [[linear order]], we speak of a [[linearly ordered group]]. |
If the order on the group is a [[linear order]], we speak of a [[linearly ordered group]]. |
Revision as of 17:40, 8 December 2007
In abstract algebra, an ordered group is a group (G,+) equipped with a partial order "≤" which 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. 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 if and only if 0 ≤ a-1 b. The set of elements x ≥ 0 of G 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+.
The order of an ordered group G is defined by G+; a group is an 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
If the order on the group is a linear order, we speak of a linearly ordered group. If the order on the group is a lattice order, we speak of a lattice 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 valuations of fields.
Examples
- A ordered vector space is an ordered group
- A Riesz space is a lattice ordered group
- A typical example of an ordered group is Zn, where the group operation is componentwise addition, and we write (a1,...,an) ≤ (b1,...,bn) if and only if ai ≤ bi (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.
References
- M. Anderson and T. Feil, Lattice Ordered Groups: an Introduction, D. Reidel, 1988.
- M. R. Darnel, The Theory of Lattice-Ordered Groups, Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, 1995.
- L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, 1963.
- A. M. W. Glass, Ordered Permutation Groups, London Math. Soc. Lecture Notes Series 55, Cambridge U. Press, 1981.
- V. M. Kopytov and A. I. Kokorin (trans. by D. Louvish), Fully Ordered Groups, Halsted Press (John Wiley & Sons), 1974.
- V. M. Kopytov and N. Ya. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, 1996.
- V. M. Kopytov and N. Ya. Medvedev, The Theory of Lattice-Ordered Groups, Mathematics and its Applications 307, Kluwer Academic Publishers, 1994.
- R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.