For Ubuntu MATE
ubuntu-mate-core
ubuntu-mate-desktop
For just the MATE Desktop:
mate-desktop-environment
It should be noted that "converting" between flavours of Ubuntu is possible, but not recommended. See here:
You'd benefit from a full retrospective experience if the system from the front and back is using MATE components, which may mean backing up and performing a clean install. As opposed to a mix and match of GNOME 2, Unity, etc which may introduce some glitches along the way.