I just tried opening a PDF with eom. It isn't a problem with your installation; it doesn't look like eom supports PDFs.
mate-desktop-environment is a metapackage that references a bunch of default applications that should be installed with mate. Since eom is one of them, removing eom would mean that not all of the applications referenced are installed, thus removing mate-desktop-environment. Not to worry though; it's just a metapackage, so you can safely remove mate-desktop-environment without doing any damage.
Why not just make the document viewer your default PDF viewer? Right click on a PDF > Properties > Open With > Select the viewer and hit "Set as default"