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As far as I understand it:
- X11 is how you draw primitive things;
- A (compositing) window manager is something that uses X11 to provide the tools for drawing more complex things, and position them in layers on screen;
- A desktop environment is something that uses a window manager to provide the bare essentials of a GUI-based operating system, like a control panel, calculator and solitaire apps, task bar, etc.
If my understanding is correct - what is Cinnamon? Its wiki article (and the tag here on SU) describe it as a "desktop environment", but I can't find what window manager Linux Mint uses by default in its wiki article, and the Tara release notes mention improving the "window manager" in the Cinnamon 3.8 section.
1Actually what "uses X11 to provide the tools for drawing more complex things" (eg: menu bars, buttons, combo boxes) is a toolkit like Tk, GTK, and Qt. As @TSJNachos117 said, a window manager decorates X11 windows with borders and titles to allow easier resizing and positioning. – Diogo Kollross – 2019-04-12T11:57:31.570
4TIL that solitaire apps is a bare essential of a GUI-based operating system. – pipe – 2019-04-12T13:01:48.830
If it doesn't have solitaire then why bother? – Adam Barnes – 2019-09-20T13:24:29.783