The local-global property for G-invariant terms

09/05/2021

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by   Alexandr Kazda, et al.

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For some Maltsev conditions Σ it is enough to check if a finite algebra 𝐀 satisfies Σ locally on subsets of bounded size, in order to decide, whether 𝐀 satisfies Σ (globally). This local-global property is the main known source of tractability results for deciding Maltsev conditions. In this paper we investigate the local-global property for the existence of a G-term, i.e. an n-ary term that is invariant under permuting its variables according to a permutation group G ≤ Sym(n). Our results imply in particular that all cyclic loop conditions (in the sense of Bodirsky, Starke, and Vucaj) have the local-global property (and thus can be decided in polynomial time), while symmetric terms of arity n>2 fail to have it.