Abstract
We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier \(\sf Q\) in \({\rm FO}({\vec Q}_k)\), the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two \({\rm FO}({\vec Q}_1)\)-equivalent models with certain additional structure, yields a pair of \({\rm FO}({\vec Q}_k)\)-equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and planarity.
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