When my system updated from kernel version 5.3.0-24-generic to 5.3.0-26-generic it sent me to the initramfs busybox prompt on re-boot instead of booting to the desktop. I have determined that the reason for this is that I use a LVM partition for my Ubuntu Mate root partition, and the new initramfs provided by the 5.3.0-26 kernel update does not seem to support LVM.
My system does still work if I select the option at boot time to boot to the old 5.3.0-24 kernel.
I would like to know if there are any other Ubuntu Mate version 19.10 users who use LVM, and have this problem. I have made a bug report on LaunchPad, https://bugs.launchpad.net/ubuntu-mate/+bug/1859097 is the URL. If anyone can confirm the bug, or let me know that your system with LVM does not have the bug, I would appreciate it.
Since no one has commented on this problem, I will update one more time and then let it go. When the initrd.img-5.3.0-29-generic became available, and my system was upgraded to it, it failed in exactly the same way that the initrd.img-5.3.0-26-generic did. For some reason neither of the initrd images after initrd.img-5.3.0-24-generic would support the LVM configuration on my computer system.
As a test, I set up a test partition in the LVM volume group and did a new install of Ubuntu-Mate version 19.10 into it. The Live boot USB stick installed initrd.img-5.3.0-18-generic, and it booted properly when I restarted the system. After running the Software Updater, which installed the update to initrd.img-5.3.0-29-generic, the system still operates properly. This indicates that something in my main system changed to prevent the initrd image from being generated properly.
Since I don't know enough about how the initrd generator works to be effective, rather than trying to figure out what happened, I decided make a list of all of the applications I installed, the customization I made to the system, and back up my user space, then regenerate the system starting with the version 19.10 USB stick. While that process was quite time consuming, it likely took less time than trying to figure out what went wrong with the system to begin with.
In any case, somewhere there was a bug that made my system unusable, but since I re-installed the system there is no way to find the problem now. So, this bug can be closed as far as I am concerned.