Given $n$ functions $f_1, f_2, ..., f_n$ (say, simple linear over $\mathbb{R}$). How many different functions can one construct from them using only $\min$ and $\max$ operations? It is clear that the answer depends on the function system itself, but is there at least some reasonably good upper estimate of this number?
Do the following considerations make sense?
Let $M_n$ be the required number, $N_n$ be the number of different functions one can construct using only $\min$ and $\max$ and using all n initial functions. Then $M_n$ is less or equal to a sum of all $N_1, N_2, ..., N_n$ taken with correspondent binomial coefficients (select $k$ unique functions from $n$)
$$M_n \leqslant {\binom n 1} N_1 + {\binom n 2}N_2 + ... + {\binom n {n-1}}N_{n-1} + N_n,$$
$$N_n \leqslant 2n!N_{n-1}$$
The last formula follows from that we can take all $n$ given functions in any order ($n!$), then we can take exactly two $\min$ and $\max$ operations over any of $N_{n-1}$ last functions.
Is it correct? Is there any more precise estimate?