Cygwin: bash shell & xterm

Singular is a package by mathematiciens for mathematiciens. If you are not a mathematicien, forget it, or better still unload it (more about that later).

When you run setup.exe of the CYGWIN package, you will eventually end up with a list of potential packages. There is a default set of such packages which will be loaded onto your computer if you do nothing. You can of course pick and choose what to load. If you are a mathematicien interested in singularities then you might choose some of the singular packages such as "Singular base" or "Singular surf". Cygwin helps with dependencies i.e. will also load any packages which are needed. I do not know Singular, but it probably has a dependency on X-Windows (for emacs perhaps) and hence loads the X-Window package and other things like X-terminal.

When cygwin starts it opens an "MS windows" window. i.e. an area of the screen managed by Windows. In there you can run ONE command line interpreter. These days most people run BASH. The window you look at is, as you say, just a terminal window. But remember it is an MS WINDOWS window. That means that you can cut and paste between that window and other "MS Windows" windows e.g. notepad.

Another option is to run an X-server. That will create another "MS Windows" window, but this time it is not a terminal but an array of pixels managed by a server. In THAT window, other programs that you run (say from the original CYGWIN window) can ask to open up an "X-Window" INSIDE the "MS Window" created by the X-server. Following so far?  As you have discovered, one of these programs is X-term. That is a program that knows NOTHING about MS-Windows and requires an X-server. It communicates with the X-server and requests an X-Window within the pixels managed by the X-server. Other programs can open other windows in that managed by the X-server (perhaps such as singular-surf). To make that useful you have also to run a "window manager" which is the program allowing you to move windows around in the pixels, close windows etc.

THEN, a shell such as BASH can be run. Bash then sends its characters to the X-term, X-terms sends it's demand to display to X-server and X-server writes them to the screen (approximate view)

So, here are two ways to end up with BASH on a windows machine. Normally this is not worth doing for a shell, normally I just use the CYGWIN window for the command line. The X-Windows version has the disadvantage that cut and paste only works inside the X-Window and NOT with other MS Windows. Some people however want to run full graphical environments (such as KDE, calendars or whatever) in the X-Window. Someone has even replaced the whole of the desktop of Windows with KDE, but that is not for the faint of heart.

If all that you are trying to achieve with CYGWIN is access to the command line facilities of BASH, then you can safely ignore X-Windows and Singular. If you wish to remove Singular, then expand the list in setup.exe (which you can run whenever you like, it is not like installing an MS-Windows application) by hitting the "view" button, scroll down to the singular packages and click on the second column from the left until you see "uninstall". Setup will remove it for you (though whether it will remove icons etc I am not sure)

That enough for now?