Solving equation systems in ω-categorical algebras

Abstract

We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an [Formula: see text]-categorical algebra [Formula: see text]. There are [Formula: see text]-categorical groups where this problem is undecidable. We show that if [Formula: see text] is an [Formula: see text]-categorical semilattice or an abelian group, then the problem is in P or NP-hard. The hard cases are precisely those where [Formula: see text] has a uniformly continuous minor-preserving map to the clone of projections on a two-element set. The results provide information about algebras [Formula: see text] such that [Formula: see text] does not satisfy this condition, and they are of independent interest in universal algebra. In our proofs we rely on the Barto–Pinsker theorem about the existence of pseudo-Siggers polymorphisms. To the best of our knowledge, this is the first time that the pseudo-Siggers identity has been used to prove a complexity dichotomy.