Partially ordered group

 ( Ordered Algebraic Structures ) 

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: if and only if

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 typeface l: ℓ-group).

A Frigyes 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 theory.

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

---

Examples

• The 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 integer ⁿ, 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 The Archimedean property of the real numbers can be generalized to partially ordered groups.

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

---

Integrally closed A partially ordered group G is called integrally closed if for all elements a and b of G, if a ⁿ ≤ 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 set group is already abelian group. This has to do with the fact that a directed group is embeddable into a complete lattice lattice-ordered group if and only if it is integrally closed.

---

See also

---

Note

• 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.
• (1982). 9780521241908 ISBN 9780521241908
• (1999). 981449609X, World Scientific. 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.
• (1994). 9789048144747 ISBN 9789048144747
• R. B. Mura and A. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
• (2026). 9781852339050 ISBN 9781852339050
  , chap. 9.

---

Further reading

---

External links

Post Comment