partially ordered group (original) (raw)

A partially ordered group is a group G that is a poset at the same time, such that if a,b∈G and a≤b, then

1. 1. a⁢c≤b⁢c, and
2. 1. c⁢a≤c⁢b,

for any c∈G. The two conditions are equivalent to the one condition c⁢a⁢d≤c⁢b⁢d for all c,d∈G. A partially ordered group is also called a po-group for short.

Remarks.

• •
  One of the immediate properties of a po-group is this: if a≤b, then b-1≤a-1. To see this, left multiply by the first inequality by a-1 on both sides to obtain e≤a-1⁢b. Then right multiply the resulting inequality on both sides by b-1 to obtain the desired inequality: b-1≤a-1.
• •
  If can be seen that for every a∈G, the automorphisms La,Ra:G→G also preserve order, and hence are order automorphisms as well. For instance, if b≤c, then La⁢(b)=a⁢b≤a⁢c=La⁢(c).
• •
• •
  (special po-groups)
  1. (a)
     A po-group whose underlying poset is a directed set is called a directed group.
     • *
       If G is a directed group, then G is also a filtered set: if a,b∈G, then there is a c∈G such that a≤c and b≤c, so that a⁢c-1⁢b≤a and a⁢c-1⁢b≤b as well.
     • *
       Also, if G is directed, then G=⟨G+⟩: for any x∈G, let a be the upper bound of {x,e} and let b=a⁢x-1. Then e≤b and x=a-1⁢b∈⟨G+⟩.
  2. (b)
     A po-group whose underlying poset is a lattice is called a lattice ordered group, or an l-group.
  3. (c)
  4. (d)
     A po-group is said to be Archimedean if an≤b for all n∈ℤ, then a=e. Equivalently, if a≠e, then for any b∈G, there is some n∈ℤ such that b<an. This is a generalization of the Archimedean property on the reals: if r∈ℝ, then there is some n∈ℕ such that r<n. To see this, pick b=r, and a=1.
  5. (e)
     A po-group is said to be integrally closed if an≤b for all n≥1, then a≤e. An integrally closed group is Archimedean: if an≤b for all n∈ℤ, then a≤e and e≤b. Since we also have (a-1)-n≤b for all n<0, this implies a-1≤e, or e≤a. Hence a=e. In fact, an directed integrally closed group is an Abelian po-group.
• •
  Since the definition above does not involve any specific group axioms, one can more generally introduce partial ordering on a semigroup in the same fashion. The result is called a partially ordered semigroup, or a po-semigroup for short. A lattice ordered semigroup is defined similarly.