Local–global property for G-invariant terms

Abstract

For some Maltsev conditions [Formula: see text] it is enough to check if a finite algebra [Formula: see text] satisfies [Formula: see text] locally on subsets of bounded size in order to decide whether [Formula: see text] satisfies [Formula: see text] (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 [Formula: see text]-ary term that is invariant under permuting its variables according to a permutation group [Formula: see text]. 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 [Formula: see text] fail to have it.