Two-element structures modulo primitive positive constructability

Primitive positive constructions have been introduced in recent work of Barto, Opršal, and Pinsker to study the computational complexity of constraint satisfaction problems. Let mathfrak P_{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_{Boole} of mathfrak P_{fin} to relational structures on a two-element set; in particular, we prove that mathfrak P_{Boole} is a lattice. Finally, we use mathfrak P_{Boole} to present the various complexity regimes of Boolean constraint satisfaction problems that were described by Allender, Bauland, Immerman, Schnoor and Vollmer.