Incidence-preserving map

Definition
Let $$S$$ and $$T$$ be two incidence systems. An incidence-preserving map from $$S$$ to $$T$$ is a map from the set of varieties of $$S$$ to the set of varieties of $$T$$ that preserves the incidence relation.

In graph-theoretic terms, it corresponds to a graph homomorphism.

For the case of an incidence structure (viz an incidence system with only two types, points and blocks), if the incidence structure is connected, then every incidence-preserving map either takes points to points or blocks to blocks, or interchanges the role of points and blocks. The incidence-preserving maps that take points to ponits are termed homomorphisms and the ones that interchange the role of points and blocks are termed antihomorphisms.