Quantifiers closed under partial polymorphisms

Abstract: We study Lindstrom quantifiers that satisfy certain closure properties which are motivated by the study of polymorphisms in the context of constraint satisfaction problems (CSP). When the algebra of polymorphisms of a finite structure B satisfies certain equations, this gives rise to a natural closure condition on the class of structures that map homomorphically to B. The collection of quantifiers that satisfy closure conditions arising from a fixed set of equations are rather more general than those arising as CSP. For any such conditions P, we define a pebble game that delimits the distinguishing power of the infinitary logic with all quantifiers that are P-closed. We use the pebble game to show that the problem of deciding whether a system of linear equations is solvable in Z2 is not expressible in the infinitary logic with all quantifiers closed under a near-unanimity condition.