Partially ordered group
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 if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and is called the positive cone of G.
By translation invariance, we have a ≤ b if and only if 0 ≤ -a + b. So we can reduce the partial order to a monadic property: a ≤ b if and only if -a + b ∈ G+.
For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially orderable 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 l: 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 x1, x2, y1, y2 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
[edit]- The integers with their usual order
- 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 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 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)
Properties
[edit]Archimedean
[edit]The Archimedean property of the real numbers can be generalized to partially ordered groups.
- Property: A partially ordered group is called Archimedean when for any , if and for all then . Equivalently, when , then for any , there is some such that .
Integrally closed
[edit]A partially ordered group G is called integrally closed if for all elements a and b of G, if an ≤ b for all natural n then a ≤ 1.[1]
This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent.[2] There is a theorem that every integrally closed directed group is already abelian. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed.[1]
See also
[edit]- Cyclically ordered group – Group with a cyclic order respected by the group operation
- Linearly ordered group – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
- Ordered field – Algebraic object with an ordered structure
- Ordered ring – ring with a compatible total order
- Ordered topological vector space
- Ordered vector space – Vector space with a partial order
- Partially ordered ring – Ring with a compatible partial order
- Partially ordered space – Partially ordered topological space
Note
[edit]References
[edit]- M. Anderson and T. Feil, Lattice Ordered Groups: an Introduction, D. Reidel, 1988.
- Birkhoff, Garrett (1942). "Lattice-Ordered Groups". The Annals of Mathematics. 43 (2): 313. doi:10.2307/1968871. ISSN 0003-486X.
- 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.
- Glass, A. M. W. (1982). Ordered Permutation Groups. doi:10.1017/CBO9780511721243. ISBN 9780521241908.
- Glass, A. M. W. (1999). Partially Ordered Groups. ISBN 981449609X.
- 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.
- Kopytov, V. M.; Medvedev, N. Ya. (1994). The Theory of Lattice-Ordered Groups. doi:10.1007/978-94-015-8304-6. ISBN 978-90-481-4474-7.
- R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
- Lattices and Ordered Algebraic Structures. Universitext. 2005. doi:10.1007/b139095. ISBN 1-85233-905-5., chap. 9.
- Elliott, George A. (1976). "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras". Journal of Algebra. 38: 29–44. doi:10.1016/0021-8693(76)90242-8.
Further reading
[edit]Everett, C. J.; Ulam, S. (1945). "On Ordered Groups". Transactions of the American Mathematical Society. 57 (2): 208–216. doi:10.2307/1990202. JSTOR 1990202.
External links
[edit]- Kopytov, V.M. (2001) [1994], "Partially ordered group", Encyclopedia of Mathematics, EMS Press
- Kopytov, V.M. (2001) [1994], "Lattice-ordered group", Encyclopedia of Mathematics, EMS Press
- This article incorporates material from partially ordered group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.