We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an ω-categorical algebra 𝔄. There are ω-categorical groups where this problem is undecidable. We show that if 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(𝔄,≠) has a uniformly continuous minor-preserving map to the clone of projections on a two-element set. The results provide information about algebras 𝔄 such that Pol(𝔄,≠) 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.

中文翻译：

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求解 ω 分类代数中的方程组

我们研究了决定一组给定的等式和不等式是否有解的计算复杂性ω- 分类代数𝔄. 有ω- 无法确定此问题的类别组。我们证明如果𝔄是一个ω-分类半格或阿贝尔群，则问题在 P 或 NP-hard 中。困难的情况正是那些波尔(𝔄,≠)有一个一致连续的次要保留映射到二元素集上的投影克隆. 结果提供了有关代数的信息𝔄这样波尔(𝔄,≠)不满足这个条件，它们对全能代数有独立的兴趣。在我们的证明中，我们依赖于关于伪 Siggers 多态性的 Barto-Pinsker 定理。据我们所知，这是第一次使用伪 Siggers 身份来证明复杂性二分法。