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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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    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.2011-06-12

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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.