Let $L$ be a finite lattice. Then $L$ is generated by its join-irreducible elements $J(L)$ or alternatively its meet-irreducible elements $M(L)$.

If $S \subseteq L$ is a sub join-semilattice then $|M(S)| \leq |M(L)|$. This follows by the self-duality of finite join-semilattices with join-preserving maps i.e. dually we have a surjective quotient $L^{op} \twoheadrightarrow S^{op}$ which implies $|J(S^{op})| \leq |J(L^{op})|$ because the join-irreducibles form the least set of generators.

However one can have $|J(S)| > |J(L)|$ e.g. take $L = \mathbf{2}^3$ and remove an atom to form $S$.

My question is whether there exist non-trivial bounds on ${max}_{S \subseteq L}(|J(S)| - |J(L)|)$ i.e. one fixes $L$ and varies over the sub join-semilattices $S$. I am mainly interested in the case $L = \mathbf{2}^n$.

In summary, how many more join-irreducibles can there be in a sub join-semilattice of a finite lattice?

Thanks for any help.