There is another thread from the past where we were covering this very issue, and needed a Mate user to verify their experience with LightDM/Mate; in regards to elogind and consolekit.  You have become that person! 

It might help if you post your success/experience with LightDM/Mate/elogind on this thread-

https://dev1galaxy.org/viewtopic.php?id=2282

Thanks!

sudo apt-get install lightdm libpolkit-backend-elogind-1-0

I think:

• lightdm should deped on libpolkit-backend-elogind-1-0

• slim should deped on libpolkit-backend-consolekit-1-0

• task-mate-desktop should depend on any of libpolkit-backend-consolekit-1-0 | libpolkit-backend-elogind-1-0

Without this last fix, installing libpolkit-backend-elogind-1-0 it's also removed task-mate-desktop and some other packages are suggested for autoremoval: gnome-orca speech-dispatcher libreoffice-gtk3 etc.

I tried this in a VM today and it worked, but you should heed the advice of the others.

Shutdown action from desktop controls is not working now.

If you are having various permission problems, you may be experiencing the infamous libpolkit problem.  Cant shut down, cant mount usb devices, cant access wifi network...

Its been my experience that LightDM requires libpolkit-backend-elogind, as opposed to libpolkit-backend-consolekit (used by slim?).  Check to see what libpolkit you have installed, I suspect consolekit?

If you have consolekit installed,  try installing the elogind polkit:

apt-get install libpolkit-backend-elogind-1-0

This will remove consolekit and install elogind.

/etc/init.d/slim stop
apt-get install lightdm

A debconf dialog will come up and you can set lightdm to be the default display manager.

Edit /etc/lightdm/lightdm.conf to uncomment and change to true this:

#greeter-hide-users=false

so it looks like this:

greeter-hide-users=true

(Make sure you're in the section under "[SeatDefaults]" and not "# Seat defaults")
Start lightdm:

/etc/init.d/lightdm start

Shutdown/reboot buttons should still work in desktop and at the login screen. I'm not sure about mounting removable drives with this combination. Some tweaking may be necessary.

Don't try to uninstall slim. It's not worth the trouble. (It'll want to pull everything with it because it was installed as part of a task package.)

If you want to change back to slim, run dpkg-reconfigure lightdm and set slim as the default.

Does anybody know if it's possible with Slim?
Otherwise, how to switch properly from Slim to Lightdm?