Local–global property for G-invariant terms

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