Satisfiability problems and algebras of boolean constraint system games

Mermin and Peres showed that there are boolean constraint systems (BCSs) which are not satisfiable, but which are satisfiable with quantum observables. This has led to a burgeoning theory of quantum satisfiability for constraint systems, connected to nonlocal games and quantum contextuality. In this theory, different types of quantum satisfying assignments can be understood as representations of the BCS algebra of the system. This theory is closely related to the theory of synchronous games and algebras, and every synchronous algebra is a BCS algebra and vice-versa. The purpose of this paper is to further develop the role of BCS algebras in this theory, and tie up some loose ends: We give a new presentation of BCS algebras in terms of joint spectral projections, and show that it is equivalent to the standard definition. We construct a constraint system which is $C^*$-satisfiable but not tracially satisfiable. We show that certain reductions between constraint systems lead to $*$-homomorphisms between the BCS algebras of the systems, and use this to streamline and strengthen several results of Atserias, Kolaitis, and Severini on analogues of Schaefer's dichotomy theorem. In particular, we show that the question of whether or not there is a non-hyperlinear group is linked to dichotomy theorems for $\mathcal{R}^{\mathcal{U}}$-satisfiability.