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We study Lindstrom quantifiers that satisfy certain closure properties whichare motivated by the study of polymorphisms in the context of constraintsatisfaction problems (CSP). When the algebra of polymorphisms of a finitestructure B satisfies certain equations, this gives rise to a natural closurecondition on the class of structures that map homomorphically to B. Thecollection of quantifiers that satisfy closure conditions arising from a fixedset of equations are rather more general than those arising as CSP. For anysuch conditions P, we define a pebble game that delimits the distinguishingpower of the infinitary logic with all quantifiers that are P-closed. We usethe pebble game to show that the problem of deciding whether a system of linearequations is solvable in Z2 is not expressible in the infinitary logic with allquantifiers closed under a near-unanimity condition.

Quantifiers closed under partial polymorphisms