Equality implies equality, name?

Asked Mar 01 '16 at 00:49

Active Mar 01 '16 at 01:09

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Let $f_1: A \rightarrow F_1$ and $f_2: A \rightarrow F_2$. How would one call the following property of these two functions:

$$\forall a_1, a_2 \in A: f_1(a_1) = f_1(a_2) \Rightarrow f_2(a_1) = f_2(a_2)$$

Does this have something to do with homomorphisms?

Would it be different if the following is also true, if $F_1$ and $F_2$ are totally ordered:

$$\forall a_1, a_2 \in A: f_1(a_1) \ge f_1(a_2) \Rightarrow f_2(a_1) \ge f_2(a_2)$$ $$\forall a_1, a_2 \in A: f_1(a_1) \le f_1(a_2) \Rightarrow f_2(a_1) \le f_2(a_2)$$

edited Mar 01 '16 at 01:09

asked Mar 01 '16 at 00:49

Marcus Thyssen

This site uses MathJax for mathematics formatting. See here for a quick and thurough guide. – Arthur Mar 01 '16 at 00:52
Ah, thanks! Modified it, so it's easier to read now. – Marcus Thyssen Mar 01 '16 at 01:13
Seems like it is a generalization of the notation of injectivity. If $f_2$ is the identity, then it is definition of injective. – yoyostein Mar 02 '16 at 06:04
(Very slightly more general, you get injectivity as soon as $f_2$ is injective.) – Clement C. Mar 02 '16 at 16:20

Equality implies equality, name?

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