Order-isomorphism and a projection’s diagram of C(X)

A mapping between projections of C ∗ -algebras preserving the orthogonality, is called an orthoisomorphism. We define the order-isomorphism mapping on C ∗ -algebras, and using Dye’s result, we prove in the case of commutative unital C ∗ -algebras that the concepts; order-isomorphism and the orthoisomorphism coincide. Also, we define the equipotence relation on the projections of C(X) ; indeed, new concepts of finiteness are introduced. The classes of projections are represented by constructing a special diagram, we study the relation between the diagram and the topological space X. We prove that an order-isomorphism, which preserves the equipotence of projections, induces a diagram-isomorphism; also if two diagrams are isomorphic, then the C ∗ -algebras are isomorphic.