For me, MATE is well installed. However, MATE don't appears in xwmconfig and there is no such things as a ~/.xinitrc file or /etc/skel/.xinitrc (like in Arch)... What shoud I do?

The package "mate-session-manager" of the MSB package set contains the file /etc/X11/xinit/xinitrc.mate-session which should make MATE a visible choice in xwmconfig. I have no idea why it won't show up for you.

The xwmconfig script will create ~/.xinitrc for you. Slackware does not have /etc/skel/.xinitrc .

Eric

@lensilvan - did you install MATE from the MATE SlackBuilds project and if so, did you use our install scripts to build the packages yourself or did you use our binary packages? Because if you installed MATE from MATE SlackBuilds then, like Eric mentioned, the xwmconfig should give you MATE as a desktop environment option.

Strangely, although set-mate was already installed, there was no mate-session-manager and I had to install it through slapt-get. Then, I find a mate entry in xwmconfig and selected it and it appeared on the ~/.xinitrc file. However, when I run startx, there was only a blank screen.

I recommend you to compile and install the packages by yourself with the sources from the Git Repo

Hmm, it sounds to me like you are not using MATE SlackBuilds or our packages since our repo is not set up for slapt-get. Do you remember where you got your Mate packages from? What repo? Are you mixing packages from different repos?

For the moment I used slapt-get but now that I have a graphical interface, I will try Slackbuilds.
In fact, many packages necessary for running slackware were missing. After I installed them, now MATE is displayed correctly, however with still some issues. Many icons are missing in the menus as you can see in the screenshot I added in my message. What should I do in this case?

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if you are not using a full installation you can meet dependency problems not only for the stuff installed but also for any third party, like MSB or the scripts from slackbuilds.org: if you are not able to resolve all dependencies yourself, you can try sticking to the only supported installation, the full one.

to add all the missing packages (you can still select to exclude what you prefer, but remember it's your choice so you have to live with the consequencies ), refresh slackpkg datas with
and then, if you have slackware (32bit) issue as root
if you have slackware64

I have installed all the missing packages, except those related to KDE. However, after rebooting, there are still menu icons missing. Also, in the system menu, I don't have the option of shutting down or reboot, only end the session and lock the screen. Also, the links to caja in the "Places" menu don't work, however I can open caja directly from the desktop...

if you're still using the salix packages and you are on slackware 14.0, try using the ones provided by chess and willy instead

http://slackware.org.uk/msb/

if you're using -current you should rebuild them yourself

https://github.com/mateslackbuilds/msb

yes, we provide binary packages for Slackware 14.0 only, not Current since Current is a moving target. Although i used Current daily, but i build the binary packages on a VM with a clean installation of Slackware 14.0 only with official updates, no other packages are installed

No problem, I use the 14 version.
After I installed the missing packages using that repo, now everything seems to work better. There are however still an issue: At each boot, I don't have in the menu the options of rebooting or shuting down.

which runlevel do you use?

Well it is set on runlevel 4...
I thought that it was only a problem of permission. I added my user to the power group. However, after rebooting the same issue occurs, incredible. This is the output of 'groups':
Is it related to MATE or something else? (In this case, I should not pollute this thread )

Can you try switch to runlevel 3 and see the result?