"Lamp-switch set-up number" of $n$ [closed]

Motivation. The following has a real-life (!) inspiration from a discussion about how to connect lamps and switches in an efficient way.

Question. Let $$n\in\mathbb{N}$$ be a positive integer and let $$\{1,\ldots,n\}$$ represent $$n$$ lamps, each of which is in exactly one of the states OFF or ON. Let $$E\subseteq {\cal P}(\{1,\ldots,n\})$$. For every $$e\in E$$ we have an "$$e$$-button", such that if that button is pressed, every element of $$e$$ switches its state (either from OFF to ON, or vice versa). We say that $$E$$ is state-complete if the following condition holds:

If all lamps are OFF and if $$k\in \{1,\ldots, n\}$$ is given, there is a finite button-sequence $$e_1, \ldots, e_m \in E$$ such that after all the buttons have been pressed, lamp $$k$$ is ON and all other lamps are OFF.

For instance, $$\big\{\{k\}: k\in \{1,\ldots,n\}\big\}$$ is state-complete, and $$\big\{\{1,\ldots,n\}\big\}$$ is not state-complete for $$n\geq 2$$.

Given $$n\in \mathbb{N}$$, what is the least cardinality that a state-complete set $$E\subseteq {\cal P}(\{1,\ldots,n\})$$ can have?

• Think of $\mathcal P(\{1,\dots,n\}$ as an $n$-dimensional vector space over the $2$-element field. You're asking for the minimal size of a set $E$ whose linear combinations include all the standard basis vectors $\{k\}$. Such an $E$ spans the whole space, so the minimum size is $n$. May 14 at 17:15
• Or, more elementary, if you have $k<n$ buttons, you get at most $2^k<2^n$ combinations of lamps, thus not all. This contradicts to the possibility to switch each separate lamp. May 14 at 17:33
• So, we have found a complicated description of the identity function. May 15 at 1:40
• hm, why do not they commute? May 15 at 8:05
• Right - I am wrong, sorry May 15 at 9:46