The category of AF $C^*$-algebras is equivalent to that of ordered abelian groups with strong unit (i.e. an element with the archimedian property, adding this one to itself often enough you can always obtain something bigger than any given element) via the $K_0$-functor. The morphisms in this category are the order preserving, unit preserving group homomorphisms. The latter category is equivalent to the category of MV-algebras via the functor plucking out the "interval" between the neutral element and the strong unit (see e.g. this book).
The category of AF $C^*$-algebras is equivalent to that of ordered abelian groups with strong unit via the $K_0$-functor. The latter is equivalent to the category of MV-algebras (see e.g. this book)