On the Jordan-Hölder decomposition of proof nets

Abstract

Having defined a notion of homology for paired graphs, Metayer ([Ma]) proves a homological correctness criterion for proof nets, and states that for any proof net $G$ there exists a Jordan-Holder decomposition of ${\mathsf H}_0(G)$ . This decomposition is determined by a certain enumeration of the pairs in $G$ . We correct his proof of this fact and show that there exists a 1-1 correspondence between these Jordan-Holder decompositions of ${\mathsf H}_0(G)$ and the possible ‘construction-orders’ of the par-net underlying $G$ .