Notation for implied sets [closed]

Let $0^X$ is the least element of a poset $X$.

Let $f$ is a function from a poset $\mathfrak{A}$ to $\mathfrak{B}$.

Consider the equation $f(0^{\mathfrak{A}}) = 0^{\mathfrak{B}}$.

Could it be written just $f(0)=0$? How to describe these kinds of notation without contradictions?

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closed as not a real question by Andres Caicedo, sdcvvc, Jason DeVito, tomasz, ThomasOct 13 '12 at 14:16

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Sure, you could write it as $f(0)=0$, but only if it's understood that $0$ always means the least element of any poset you're considering.
As an example of how it could be helpful to use this special notation, consider the inclusion of posets $f:[0,1]\hookrightarrow [-1,1]$ (closed intervals of the real line). You could say that $f(0)=0$, using $0$ to denote an ordinary real number, but that $f(0^{[0,1]})\neq 0^{[-1,1]}$, since $f$ does not send the least element of $[0,1]$ to the least element of $[-1,1]$.