There's a sequence of natural numbers as follows:
Definition. Given $n \in \mathbb{N}$, define that $a_n$ is the number of "good" subsets of the free distributive lattice on $n$ generators.
The term "good" is defined as follows: given a lattice $L$ with a subset $S$, lets call $S$ big iff for all lattice homomorphisms $f :X \rightarrow \{0,1\}$, it holds that if $f$ is constant when restricted to $S$, then $f$ is indeed a constant function. Call $S$ good iff it is big, and minimal as such.
Question. Does this sequence (or any variant thereof) have a name, and/or has anyone studied it?
Here's why I find this sequence interesting.
If two booleans $A$ and $B$ are equal, then so too are all of: $$A,B, A\wedge B, A \vee B.$$ This corresponds to the fact that $\{a,b\}$ is a "big" subset of the free lattice on $\{a,b\}$. Indeed, it's minimal among big subsets, ergo good. But we can also deduce that $A=B$ from the condition $A \vee B = A \wedge B$. This corresponds to the fact that $\{a \wedge b, a \vee b\}$ is a big subset of the free lattice on $\{a,b\}$, and indeed, by minimality, its a good subset.
More generally, if we have a set of booleans, it might be useful to know which subsets of the lattice generated by those booleans need to be demonstrated to be singletons before the entire set of booleans can be deemed a singleton. And of course, we want to minimize the amount of work for ourselves, so we're searching for minimal such subsets of our collection of booleans. The sequence above is defined so if we have $n$ booleans, the number of minimal such subsets is precisely $a_n$. I haven't actually proven this, by the way - it's just an intuition.