Abstract — We study a two-transmitter two-receiver network where Receiver 1 can only hear the transmitted signal of Transmitter 1. Transmitter 1 has two messages, one of which is intended for Receiver 1 while both are intended for Receiver 2. Transmitter 2 has one message which is intended for Receiver 2. We call this channel model the Z channel with degraded message sets. When the multiple access link between the two transmitters and Receiver 2 satisfies certain conditions, we characterize the capacity region of the Z channel with degraded message sets despite the presence of distributed encoding.