Name for a function preserved over a relation

In summary, the special name for f in this case would be "order preserving" when the relation R is an inequality. For other relations, it may be referred to as R-preserving or possibly an endomorphism.
  • #1
Tac-Tics
816
7
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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  • #2
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
 
  • #3
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.
 

FAQ: Name for a function preserved over a relation

What is a function preserved over a relation?

A function preserved over a relation is a mathematical concept in which a certain relationship between two sets of elements is maintained or "preserved" when a function is applied to those elements. This means that the output of the function will still maintain the same relationship with the elements in the second set as the original elements did with the first set.

What are some common examples of functions preserved over a relation?

Some common examples of functions preserved over a relation include addition, subtraction, multiplication, division, and exponentiation. These functions maintain the same relationship between the elements in the two sets, such as the relationship of "twice as much" or "half as much".

How is a function preserved over a relation different from a regular function?

A function preserved over a relation differs from a regular function in that it maintains a specific relationship between the elements in two sets, rather than simply mapping one set of elements to another. This means that the output of the function will still have a meaningful relationship with the elements in the second set, while a regular function may not have any specific relationship between its input and output.

What are the practical applications of functions preserved over a relation?

Functions preserved over a relation have many practical applications in fields such as computer science, physics, and economics. They can be used to model and analyze relationships between variables, make predictions, and solve complex problems.

How can I identify if a function is preserved over a relation?

To identify if a function is preserved over a relation, you can first determine the relationship between the elements in the two sets. Then, apply the function to the elements and see if the output maintains the same relationship with the elements in the second set. If it does, then the function is preserved over the relation.

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