Partially ordered group

Short description: Group with a compatible partial order

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 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.

Partially ordered groups are used in the definition of valuations of fields.

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

Archimedean

Archimedean property of the real numbers can be generalized to partially ordered groups.

Property: A partially ordered group [math]\displaystyle{ G }[/math] is called Archimedean when for any [math]\displaystyle{ a, b \in G }[/math], if [math]\displaystyle{ e \le a \le b }[/math] and [math]\displaystyle{ a^n \le b }[/math] for all [math]\displaystyle{ n \ge 1 }[/math] then [math]\displaystyle{ a=e }[/math]. Equivalently, when [math]\displaystyle{ a \neq e }[/math], then for any [math]\displaystyle{ b \in G }[/math], there is some [math]\displaystyle{ n\in \mathbb{Z} }[/math] such that [math]\displaystyle{ b \lt a^n }[/math].

Integrally closed

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.

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. 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.

See also

• 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
• 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

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. http://dx.doi.org/10.2307/1968871. 
• 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 9780521241908. 
• Glass, A. M. W. (1999). Partially Ordered Groups. ISBN 981449609X. https://books.google.com/books?id=5oTVCgAAQBAJ&pg=PA191. 
• 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

Everett, C. J.; Ulam, S. (1945). "On Ordered Groups". Transactions of the American Mathematical Society 57 (2): 208–216. doi:10.2307/1990202. 

External links

• Hazewinkel, Michiel, ed. (2001), "Partially ordered group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Hazewinkel, Michiel, ed. (2001), "Lattice-ordered group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• This article incorporates material from partially ordered group on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.