For several Maltsev conditions (e.g. the existence of a Maltsev term, majority term, or cyclic term of a fixed arity) it suffices to check if they hold 'locally' (i.e. restricted to every tuple) in order to determine if they are satisfied 'globally' in a finite idempotent algebra. Using this 'local-to-global' property is one of the few known methods to show the tractability of checking a fixed Maltsev condition.
In this talk, I would like to discuss the local-to-global approach for terms, whose variables are invariant under permutations from a given permutation group . In several cases, including symmetric terms of arity (so is the symmetric group) the local-to-global approach fails.