## AbstractWe contribute to the algebraic study of the complexity of constraint satisfaction problems. We give a new sufficient condition on a set of relations $\Gamma$ over a domain $S$ for the tractability of $\csp(\Gamma)$: if~$S$ is a block-group (a particular class of semigroups) of exponent $\omega$ and $\Gamma$ is a set of relations over $S$ preserved by the operation defined by the polynomial $f(x,y,z) = xy^{\omega -1}z$ over $S$, then $\csp(\Gamma)$ is tractable. This theorem strictly improves on results of Feder and Vardi and Bulatov et al.\ and we demonstrate it by reproving an upper bound of Kl{\'\i }ma et al.\ We also investigate systematically the tractability of $\csp(\Gamma)$ when $\Gamma$ is a set of relations closed under operations that are all expressible as polynomials over a finite semigroup $S$. In particular, if $S$ is a nilpotent group, we show that $\csp(\Gamma)$ is tractable iff one of these polynomials defines a Malt'sev operation, and conjecture that this holds for all groups.
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