In projective geometry, what do we need to show an isomorphism(or that two geometries are the same)

Question

We have X(set), I(type), and *(incidence relation). I believe my teacher said we need to only check that we have a bijection in terms of X(elements) and I(type), but what about the incidence relation?

Answer 1

Reading Ueberberg chapter 2 section 2:

• A morphism requires preservation of incidences and type equalities. So incident elements must map to incident elements, and elements of the same kind must map to elements of the same kind, although that resulting kind might be different from the original one.
• A homomorphism additionally requires preservation of types, so elements of a given kind must map to elements of the same kind as the one they started from.
• An isomorphism requires a bijective homomorphism, whose inverse is a homomorphism as well. Which unless I missed something mostly means that no non-incident objects may become incident, in addition to the above requirements.

So since each definition builds on the one before, and incidence preservation is at the very beginning, you do have to preserve that as well. Could it be you misunderstood your teacher?