On classifying monotone complete algebras of operators

We give a classification of “small” monotone complete C ^*-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2 ^c distinct equivalence classes (where c is the cardinality of the continuum). This remains true when the classification is restricted to special classes of monotone complete C ^*-algebras e.g. factors, injective factors, injective operator systems and commutative algebras which are subalgebras of ℓ^∞. Some examples and applications are given.