Extension Preservation Theorems on Classes of Acyclic Finite Structures

A class of structures satisfies the extension preservation theorem if, on this class, every first order sentence is preserved under extension iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs (γ, β and α-acyclicity and also acyclicity on hypergraph quotients) and estimate their influence on the validity of the extension preservation theorem on classes of finite structures. More precisely, we prove that γ-acyclic classes (with some closure properties) satisfy the extension preservation theorem, whereas β-acyclic classes do not. We also extend the validity of the extension preservation theorem for a generalization of γacyclicity that we call γ-acyclic k-quotient. To achieve this, we make a reduction from finite structures to their k-quotients and we use combinatorial arguments on γ-acyclic hypergraphs.