Partially ordered group: Difference between revisions

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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,...,a_n) ≤ (b₁,...,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 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.

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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 ''ag'' ≤ ''bg'' and ''ga'' ≤ ''gb''. 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''.
+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''.
-By the definition, we can reduce the partial order to a monadic property: ''a'' ≤ ''b'' if and only if ''1'' ≤ ''a''<sup>-1</sup> ''b''. The set of elements ''x'' ≥ ''1'' 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''<sup>-1</sup>''b'' ∈ ''G''<sup>+</sup>.
+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>.
 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:
-* ''1'' ∈ ''H''
+* ''0'' ∈ ''H''
-* if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''ab'' ∈ ''H''
+* if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a+b'' ∈ ''H''
-* if ''a'' ∈ ''H'' then ''x''<sup>-1</sup>''ax'' ∈ ''H'' for each ''x'' of ''G''
+* if ''a'' ∈ ''H'' then ''-x''+''a''+''x'' ∈ ''H'' for each ''x'' of ''G''
-* if ''a'' ∈ ''H'' and ''a''<sup>-1</sup> ∈ ''H'' then ''a=1''
+* 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]].