Order-preserving maps

Question

If I understand correctly, order-preserving maps are generally defined between reflexive, transitive, anti-symmetric relations (Posets). Does it make sense to talk about "order-preserving maps" between reflexive and transitive relations (Quasi-orderings)? Is there an equivalent term?

Answer 1

Quasi-order preserving is fine.

Note that it makes sense to talk about a relation-preserving map between two sets endowed with arbitrary relations. That is, a map $f:A\to B$ such that $$\forall x,y\in A, \ \left[x\mathcal{R}_A y\implies f(x)\mathcal{R}_B f(y)\right]$$