Normal subsystems of fusion systems and partial normal subgroups of localities

Linking systems were introduced to study classifying spaces of fusion systems. Every linking system over a saturated fusion system F corresponds to a group-like structure called a locality. Given such a locality L, we prove that there is a one-to-one correspondence between the partial normal subgroups of L and the normal subsystems of the fusion system F. As a corollary we obtain that, for any two normal subsystems of a saturated fusion system, there exists a normal subsystem which plays the role of a "product".