Well, at least for the Logitech Elite Duo, no, since they both use a single transmitter/base. Dunno if the MX has it's own base tho. Logitech does make that keyboard separatly in a corded version, the logitech elite.
If they sell it alone, yes. Otherwise, no. They use the same base, so you'd have to use both. Honestly, it's a great mouse -- I moved from corded to cordless with no issues at all - great mouse AND it recharges itself on the base station. May as well just get both.
I use a track ball to aticked to it I love not needing a surface just for teh mouse. I can put teh Trackball on my chest lean back and do some casual gaming and surfing with no real problem .
I just need a new keyboard because i spilled coke on my old one RIP. Than my reserve has this crapy half sized shift key that is a pain. So i wanna go for a good keyboard now
EDIT: The corded elite keyboard. What are other good cordless Boards?