# What are functions applied to either side of a relation that maintain the relation called?

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.

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If you have an equivalence relation $\sim$, a function or operator $f$ is said to "be well defined modulo $\sim$" if $x\sim y$ implies $f(x)\sim f(y)$. Equality is a particularly bad example because every function or process must respect =. –  Arturo Magidin Jun 12 '11 at 4:01
Please try to make your posts self-contained; the question should be posed in the body, and you should not relay on the title to begin your presentation. –  Arturo Magidin Jun 12 '11 at 4:02

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. $x \sim y \implies f(x) \sim f(y)$.