Polynomially Closed Co-clones

Two well-studied closure operators for relations are based on primitive positive (p.p.) definitions and quantifier free p.p. definitions. The latter do however have limited expressiveness and the corresponding lattice of strong partial clones is uncountable. We consider implementations allowing polynomially many existentially quantified variables and obtain a dichotomy for co-clones where such implementations are enough to implement any relation and prove (1) that all remaining co-clones contain relations requiring a superpolynomial amount of quantified variables and (2) that the strong partial clones corresponding to two of these co-clones are of infinite order whenever the set of invariant relations can be finitely generated.