Ordering finite variable types with generalized quantifiers

Let Q be a finite set of generalized quantifiers. By L^k(Q) we denote the k-variable fragment of FO(Q), first order logic extended with Q. We show that for each k, there is a PFP(Q)-definable linear pre-order whose equivalence classes in any finite structure 21 are the L^k(Q)-types in 21. For some special classes of generalized quantifiers Q, we show that such an ordering of L^k(Q)-types is already definable in IFP(Q). As applications of the above results, we prove some generalizations of the Abiteboul-Vianu theorem. For instance, we show that for any finite set Q of modular counting quantifiers, P=PSPACE if, and only if, IFP(Q)=PFP(Q) over finite structures. On the other hand, we show that an ordering of L ^k(Q)-types is not always definable in IFP(Q). Indeed, we construct a single, polynomial time computable quantifier P such that the equivalence relation ≡^k,P, and hence ordering on L ^k(P)-types, is not definable in IFP(P)