Partially Ordered Group - Examples

Examples

  • 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,...,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 then K0(A) is a partially ordered abelian group. (Elliott, 1976)

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