Preservation and definability for the fluted fragment

Yiwen Ding

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arXiv · 2607.12970

Preservation and definability for the fluted fragment

Yiwen Ding

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Abstract

This paper investigates preservation and definability for the equality-free fluted fragment over finite relational signatures. We first prove an equirank homomorphism preservation theorem: every homomorphism-preserved fluted sentence has an existential-positive fluted equivalent of no greater quantifier rank. We then refute Purdy's claimed Los--Tarski preservation theorem for the fluted fragment. Our counterexample is a rank-three sentence over one binary relation that is preserved under extensions but has no existential fluted equivalent. Finally, a two-variable counterexample over a unary base signature shows that weak and ordinary Beth definability both fail. All three results hold over all structures and over finite structures.

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Yiwen Ding. 2026-07-14. Preservation and definability for the fluted fragment. https://arxiv.org/abs/2607.12970

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