Finite partial prime-ideal diagrams

Let $B$ be a Boolean algebra and let $A\subseteq B$ be a finite Boolean subalgebra. A finite partial prime-ideal diagram on $A$ is a Boolean homomorphism

$e:A⟶\mathbf{2}=\{0,1\}\mathrm{.}$

Thus $e(0)=0$, $e(1)=1$, and $e$ preserves complements, binary meets, and binary joins. Its partial ideal side is $e^{-1}(0)$. It is a proper prime ideal of $A$: preservation gives ideal closure, while $e(a\wedge b)=0$ in $\mathbf{2}$ implies $e(a)=0$ or $e(b)=0$.

For a finite list $F=(b_0,\dots ,b_{m-1})$ from $B$, write $b_j^1=b_j$ and $b_j^0=\mathrm{\neg }{b}_j$. The nonzero cells

$c_{\epsilon }={\bigwedge }_{j<m}{b}_j^{\epsilon (j)}(\epsilon \in {\mathbf{2}}^m)$

are precisely the atoms of the finite subalgebra $⟨F⟩$ generated by $F$; every member of $⟨F⟩$ is a join of some of these finitely many cells. Repetitions in $F$, zero cells, and the empty list cause no ambiguity: zero cells are discarded, and $⟨\mathrm{\varnothing }⟩=\{0,1\}$.

A diagram decides a finite subset $F\subseteq B$ when its domain is the whole finite subalgebra $⟨F⟩$. Requiring a homomorphism on that whole domain records every Boolean consequence among the elements of $F$; an arbitrary truth assignment merely consistent with some displayed equations is not a partial prime-ideal diagram in this sense.