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Revisiting the generalized {\L}o\'s-Tarski theorem
We present a new proof of the generalized {\L}o\'s-Tarski theorem
() introduced in [1], over arbitrary structures. Instead of
using -saturation as in [1], we construct just the "required
saturation" directly using ascending chains of structures. We also strengthen
the failure of in the finite shown in [2], by strengthening
the failure of the {\L}o\'s-Tarski theorem in this context. In particular, we
prove that not just universal sentences, but for each fixed , even
sentences containing existential quantifiers fail to capture
hereditariness in the finite. We conclude with two problems as future
directions, concerning the {\L}o\'s-Tarski theorem and , both
in the context of all finite structures.
[1] 10.1016/j.apal.2015.11.001 ; [2] 10.1007/978-3-642-32621-9\_22Comment: 12 page