Hypergraph Acyclicity and Extension Preservation Theorems

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 (gamma, beta and alpha-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 gamma-acyclic classes satisfy the extension preservation theorem, whereas beta-acyclic classes do not. We also extend the validity of the extension preservation theorem for a generalization of gamma-acyclicity that we call gamma-acyclic k-quotient. To achieve this, we make a reduction from finite structures to their k-quotients and we use combinatorial arguments on gamma-acyclic hypergraphs.