  Primitive positive constructions have been introduced in recent work of
Barto, Opr\v{s}al, and Pinsker to study the computational complexity of
constraint satisfaction problems. Let $\mathfrak P_{\operatorname{fin}}$ be the
poset which arises from ordering all finite relational structures by
pp-constructability. This poset is infinite, but we do not know whether it is
uncountable. In this paper, we give a complete description of the restriction
$\mathfrak P_{\operatorname{Boole}}$ of $\mathfrak P_{\operatorname{fin}}$ to
relational structures on a two-element set; in particular, we prove that
$\mathfrak P_{\operatorname{Boole}}$ is a lattice. Finally, we use $\mathfrak
P_{\operatorname{Boole}}$ to present the various complexity regimes of Boolean
constraint satisfaction problems that were described by Allender, Bauland,
Immerman, Schnoor and Vollmer.
