Quantifiers and Congruence Closure

We prove some results about the limitations of the expressive power of quantiiers on nite structures. We deene the concept of a bounded quantiier and prove that every relativizing quantiier which is bounded is already rst-order deenable (Theorem 3.8). We weaken the concept of congruence closed (see 6]) to weakly congruence closed by restricting to congruence relations where all classes have the same size. Adapting the concept of a thin quantiier (Caicedo 1]) to the framework of nite structures, we deene the concept of a meager quantiier. We show that no proper extension of rst-order logic by means of meager quantiiers is weakly congruence closed (Theorem 4.9). We prove the failure of the full congruence closure property for logics which extend rst-order logic by means of meager quantiiers, arbitrary monadic quantiiers, and the HH artig quantiier (Theorem 6.1).