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).

By Holland's embedding theorem, a representation theorem of Wilbur Charles Holland from 1963, every lattice-ordered group embeds, with its lattice operations preserved, into the lattice-ordered group of order-preserving bijections of a chain.

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 x_i ≤ y_j, then there exists z ∈ G such that x_i ≤ z ≤ y_j.

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.

Examples

• 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,...,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 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

See also

• Cyclically ordered group – Group with a cyclic order respected by the group operation
• Linearly ordered group – Group with translationally invariant total order
• Ordered field – Algebraic object with an ordered structure
• Ordered ring
• 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

References

• 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. JSTOR 1968871.
• 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 978-0-521-24190-8.
• Glass, A. M. W. (1999). Partially Ordered Groups. World Scientific. 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.
• Blyth, T.S. (2005). Lattices and Ordered Algebraic Structures. Universitext. 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

• 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

• 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.