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.

翻译：我们研究满足特定闭包性质的Lindström量化词，这些性质源于约束满足问题（CSP）中对多态性的研究。当有限结构B的多态代数满足某些方程时，这会在同态映射到B的结构类上自然产生一种闭包条件。由固定方程组产生的闭包条件所对应的量化词集合，比CSP所产生的量化词更具一般性。对于任意此类条件P，我们定义了一种博弈框架，该框架界定了所有P-封闭量化词所对应的无穷逻辑的区分能力。利用该博弈，我们证明了判定线性方程组在Z2中是否可解的问题，无法用所有满足近一致条件的封闭量化词所对应的无穷逻辑来表达。