I am looking for all the sub-monoids of $(\mathbb Z_2^n, *)$, i.e., binary vectors over bitwise multiplication that induce an equivalence relation as described in Characterize kernels of monoid homomorphisms.
So far, I have only been able to find the following sets:
$$M_{\mathscr{S}} = \{x : x_i = 1, i \in \mathscr{S}, x \in \mathbb{Z}_2^n\}, $$
that form submonoids over bitwise multiplication. These are the submonoids constructed by restricting a subset $\mathscr{S}$ of the coordinates to $1$. These also happen to satisfy the conditions in the above linked question.
Are there any other submonoids that induce an equivalence relation?