What are functions applied to either side of a relation that maintain the relation called? What are these kinds of processes/functions/operations called? What are the functions that violate the relation called?
Edit : removing the comment about equality relation.
I think it would be acceptable and within convention to say the function is "relation-preserving" because the relation still holds true under the action of the function, i.e. $x \sim y \implies f(x) \sim f(y) $.
Mathematicians often speak of certain properties or even equations being preserved under maps and transformations, so this choice of term has support.