Sadly, since Ubuntu is no longer considering 32-bit architecture something worth maintaining you may be better off using Ubuntu MATE as a learning platform for navigating the MATE DE, then move to Debian proper which still support 32-bit, or leaving the Debian userland using solutions like CentOS, Slackware, Gentoo etc.
I would not recommend trying to mix-and-match things unless you want to invest hours of time into making a hybrid Debian / Ubuntu installation work, but if you insist on using Ubuntu MATE you could punch this into the terminal and see if your 32-bit machine will boot;
In x-terminal-emulator:
sudo add-architecture i386
sudo apt update
Once that is done, you should be able to install a 32-bit capable Linux kernel and boot into it. However you will need to install 32-bit libraries for existing software to function on a 32-bit system, and with the exclusion of multiarch-support you may have to fetch that from Debian for the latest copy available from here.
Honestly this sounds like something I could have a play with when I have nothing better to do.
Getting back to the original thought of the post ( Why you should upgrade Windows 7 to Ubuntu MATE) for any outsiders that might stumble on here, this forum itself is one reason to use Ubuntu Mate. The willingness of people here to help others solve their problem is amazing. If you have ever visited the Debian forum or the PC Linux forum and seen the abusive, snide, critical comments towards newbies you will understand. I have never seen RTFM (read the f**king manual) here. I have seen one other forum this helpful and that would be Zorin, which I used early on, but did not like their changes. Not only are people here happy to help with Ubuntu Mate problems, they are willing to help with other software problems that have nothing to do with Ubuntu Mate. Kudos to all those people helping others.
Thank you for your reply to my query. Since I am a competent computer user, almost always Windows, but not a computer programmer (geek?), much, if not most, of the Linux/Ubuntu jargon, acronyms, etc., is Greek to me. I really get lost trying to follow it. When you speak of MATE DE, Debian, CentOS, Slackware, Gentoo, etc., I don’t know the significance of these things. Sorry about that but I really need more detail to understand your effort to help. I find that the use weird names, acronyms, etc., is a problem with Linux/Ubuntu. For example: a Document Viewer is called ATRIL; a File Browser & Manager is a CAJA; a Photo Organizer is a SHOTWELL; a Text Editor is named PLUMA; and on and on............ With Windows, the utilities names usually indicate or describe their function, e.g. “File Manager”.
I think it is a mistake for Ubuntu to no longer support 32-bit computers because, like myself, I am sure that many persons save their older machines by switching from Windows to Ubuntu MATE when their Microsoft OS is no longer supported. I would bet that most of these machines are 32-bit. If Ubuntu will not support this architecture, no use switching to it.
Google truly is your friend, and a more reliable entity than I. The only reason I mentioned the other operating systems is because they all most likely have MATE varieties (officially- or community-maintained) and support 32-bit out of the box.
I find it just as regrettable 32-bit support is going away, While Ubuntu is Debian at its core (hence the need for .deb packages to install software) it is different enough to where you can't just have sources for Ubuntu and Debian concurrently in /etc/apt/sources.list.d with expectation they'll play nice enough to not cause system instability for the sake of having a modern-day 32-bit Ubuntu.
There are other concerns about 32-bit — the fact that newer motherboards support memory far beyond the capacity of 32-bit counterparts, and the maturity of AMD64 architecture. While 32-bit will never die in our hearts, its time has long come to pass and keeping 32-bit systems alive are going to be more of a challenge on mainstream platforms as even Arch and Fedora — two other mainstream distributions available in multiple varieties like Ubuntu — had dropped 32-bit support in favour for smaller ISOs and less complexity.
About names of programs, don't get hung up on them. You are more-than-able to search by generic name in Brisk Menu and MATE Menu, and if you really have to for sake of operating in the terminal create / edit $HOME/.bash-aliases to put a bunch of alternative names for programs. For the time being just use the platform as a learning experience for trying other desktop environments in the future, or if your mind is made up find 32-bit distributions with MATE pre-installed so you don't have to tinker too much before beginning to use a new operating system using the Linux kernel.
In