AI 中文总结
研究槽形片段的洛斯 - 塔尔斯基保持定理是否成立，通过在特定无等式词汇表上构造量词秩为三的槽形句子，反驳了Purdy的相关结论，证明该定理在此不成立。
AI 中文摘要
经典的洛斯 - 塔尔斯基定理将在扩张下保持的一阶句子刻画为存在可定义的句子。在[6]中，Purdy声称类似的保持定理对槽形片段成立。我们通过在仅含一个二元关系符号的无等式词汇表上构造一个量词秩为三的槽形句子来反驳这一说法，该句子在扩张下保持，但即使在有限结构上也不等价于任何存在性槽形句子。
英文摘要
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.
Comments25 pages, 1 table. Substantially expanded version of v1: adds an equirank homomorphism preservation theorem and Beth-definability counterexamples, and adopts the LMCS style
