As stated by Wikipedia:
A sufficient statistic is minimal sufficient if it can be represented as a function of any other sufficient statistic. In other words, $S(X)$ is minimal sufficient if and only if $S(X)$ is sufficient, and if $T(X)$ is sufficient, then there exists a function f such that $S(X) = f(T(X))$. Intuitively, a minimal sufficient statistic most efficiently captures all possible information about the parameter $\theta$.
I have some trouble understanding the full meaning of "minimal"
What I don't get is, what happens if we have a third sufficient statistic, that we call $U(X)$ such that:
In this case are both $U(X)$ and $S(X)$ minimal? I ask this because minimal makes me think that it must be the "most minimal" so there can be only one (group) of minimal statistics.
If I am not wrong if f or g are invertible Then the three sufficient statistics are all minimal but in this case they all belong to the same group. In the case f and g are not invertible,the three statistics: $U(X),S(X),T(X)$ have all different efficiency in capturing information