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Partially ordered group

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(Redirected from Partially ordered monoid) Group with a compatible partial order "Ordered group" redirects here. For groups with a total or linear order, see Linearly 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

  • 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 Z, 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

Archimedean

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

Property: A partially ordered group G {\displaystyle G} is called Archimedean when for any a , b ∈ G {\displaystyle a,b\in G} , if e ≤ a ≤ b {\displaystyle e\leq a\leq b} and a n ≤ b {\displaystyle a^{n}\leq b} for all n ≥ 1 {\displaystyle n\geq 1} then a = e {\displaystyle a=e} . Equivalently, when a ≠ e {\displaystyle a\neq e} , then for any b ∈ G {\displaystyle b\in G} , there is some n ∈ Z {\displaystyle n\in \mathbb {Z} } such that b < a n {\displaystyle b<a^{n}} .

Integrally closed

A partially ordered group G is called integrally closed if for all elements a and b of G, if a ≤ 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

Note

  1. ^ Glass (1999)
  2. Birkhoff (1942)

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 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

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.

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