An order unit space is a partially ordered vector space over the field of the reals along with the choice of a particular vector called the order unit. It must satisfy:

  • Order unit property: Every vector in the space is "less than or equal to" a real multiple of the order unit.
  • Archimedean Property: If a vector is "less than or equal to" every positive real multiple of the order unit, then it is less than 0.

I will give two examples of order unit spaces. The complex numbers can be seen as a two-dimensional vector space, with basis {1,i}. We can take, say, 1 as the order unit, and say that vectors are ordered according to their real parts. Of course, a total ordering is impossible

Another example would be the 2x2 self-adjoint matrices, with I as the order unit. Then we can say that one matrix is "less than" another if its lesser eigenvalue is less than the other's lesser eigenvalue.

Source: Alfsen, E. M., Compact convex sets and boundary intervals, Springer-Verlag, Berlin, New York, 1971.
