Preservation under Substructures modulo Bounded Cores
Abstract
We investigate a model-theoretic property that generalizes the classical notion of "preservation under substructures". We call this property \emph{preservation under substructures modulo bounded cores}, and present a syntactic characterization via sentences for properties of arbitrary structures definable by FO sentences. As a sharper characterization, we further show that the count of existential quantifiers in the sentence equals the size of the smallest bounded core. We also present our results on the sharper characterization for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the sharper characterization fails, but for which the classical {\L}o\'s-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the {\L}o\'s-Tarski theorem for some of the aforementioned cases.
Cite
@article{arxiv.1205.1358,
title = {Preservation under Substructures modulo Bounded Cores},
author = {Abhisekh Sankaran and Bharat Adsul and Vivek Madan and Pritish Kamath and Supratik Chakraborty},
journal= {arXiv preprint arXiv:1205.1358},
year = {2012}
}
Comments
From v2 to v3: Corrected typos, edited sentences for better readability; Conjecture 1 of v2 is now resolved so it is now Theorem 4, its proof is included in a new section (Section 7), Thm i in v2 is now Thm i+1 for i >= 4; everything else remains the same. From v1 to v2: Thm i is now Thm i-1 for i >= 7, Corrected the proof of Theorem 10 (now Theorem 9) for B > 2 (statement is still correct)