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 $ω$-categorical algebra $\mathfrak{A}$. There are $ω$-categorical groups where this problem is undecidable. We show that if $\mathfrak{A}$ is an $ω$-categorical semilattice or an abelian group, then the problem is in P or NP-hard. The hard cases are precisely those where Pol$(\mathfrak{A},\neq)$ has a uniformly continuous minor-preserving map to the clone of projections on a two-element set. The results provide information about algebras $\mathfrak{A}$ such that Pol$(\mathfrak{A},\neq)$ 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.