Partially ordered group (original) (raw)

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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:

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 _x_1, _x_2, _y_1, _y_2 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.

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

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

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.[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]

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

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.