Comparison Properties of the Cuntz semigroup and applications to C*-algebras

We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{o}rdams simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.