Measurement-induced multipartite entanglement for distant four-level atoms in Markovian and non-Markovian environments

•The analytical results for the non-Markovian dynamics of entanglement is showed.•The atom and cavity field finally reach an entangled steady state.•The long-living maximal multipartite entanglement of distant atoms is generated.•The generation scheme can be implemented by using linear optical elements.•The maximal entanglement can be generated by adjusting the system parameters.