How to extract the filename without the extension from a full path?

The usual way to do this in bash is to use parameter expansion. (See the bash man page and search for "Parameter Expansion".)

a=${1%.*}

The % indicates that everything matching the pattern following (.*) from the right, using the shortest match possible, is to be deleted from the parameter $1. In this case, you don't need double-quotes (") around the expression.

If you know the extension, you can use basename

$ basename /home/jsmith/base.wiki .wiki
base