Name for a function preserved over a relation

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The discussion centers on the classification of functions that preserve a relation R, specifically when the implication x R y leads to f(x) R f(y). Functions that maintain simple equality fall under this category, while positive-scaling functions like f(x) = ax (for positive a) preserve the relation <. The term "order preserving" is used for functions that maintain inequalities, but a general term for R-preserving functions is not established. The term "endomorphism" may apply if the function maps a set S to itself, but clarity on terminology is needed.

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Given a relation R and a function f, is there a special name for f when

x R y implies f(x) R f(y)?

For example, if the relation R is simple equality, then all functions are of this type.

If R is <, then positive-scaling functions f(x) = ax (for positive a) are of this type.

A non-example would be f(x) = -x when the relation is <, because

"1 < 2 implies -1 < -2" is a false statement.
 
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In the case of inequalities, f is called order preserving. I don't think there's a general name for when it preserves a relation R though... probably just say it's R-preserving, define that once and be done with it. Everyone will understand what you mean
 
I'm not sure if the term "homomorphism" would apply here. Or other ____morphism term (fill in the blanks).

By placing R (the same R) on both sides of the implication, you seem to suggest than f is a function from some set S to the same set S, in which case the term "endomorphism" might apply. But I'm not sure.
 

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