I've stuck when I installed the Mate package , not only I can't update my system more but also I can't open the software center and synaptic packet manager !
How can I remove the mate with all its details from my ubuntu machine ?
If you have install MATE via the linux mint repository, then you will have upgraded many of the core ubuntu packages. You will find it very difficult to "remove" MATE after this sort of upgrade. I would suggest a restore from backup or just a complete reinstall...
However - if your concern is that software center/synaptic is not working then you can fix this as per the linked question.
N.B. IMHO the better way to experiment with MATE is per my answer here: How to install MATE in 11.10 and above