What Is The Name Of This Type Of Lock?

Mortis lock?

Half mortise lock.

https://www.horton-brasses.com/furniture-locks.asp

You should find what you are looking for at Horton.  Good luck.

Also try:   https://whitechapel-ltd.com/type/locks.html?product_list_limit=36

Thank you very much guys!

@Oldtool  Wow, both of those woodworking supply companies have a great selection of those type of locks.
But now that I know the exact search term I can shop for the best deal.  Thanks again!

Most of them are upwards of $25 but here is a cheap half mortise lock:
https://www.ebay.com/itm/272763715013?hash=item3f81fbfdc5:g:CDQAAOSwb4Vbn~ho&amdata=enc%3AAQAIAAAAwOGp0LnosGWP3Ab9kmpbgttavIRwJ6RwORejT4cmxoqDmHUUy6X9SkzNHOomOJ9lVGWGIFUzv2AftQHBLE8rBs00nlIKi5S%2BEpH8%2BZqQ5lzQRETUFmunwhmwkrkHugZtBgMDpF3gRTeTMGFQ%2F5qCkJR8AQ1yNGK06wodEfxzpIFycsN4scKPKDx2I2cMExNBlcDvp2gooSuGQrets73LzCqa%2BEtDBFaI7ltQjuexW12bRJYYfefK%2Fo4XrPKniHNNMA%3D%3D%7Ctkp%3ABk9SR_CkuMO-Yg

Thanks Jim, no doubt that is the cheapest I've seen for a half mortise lock.  Polished steel though and I've been gathering brass fittings......but I'm going to keep that in mind and I appreciate you finding that!  Seems like everything gives me sticker shock...lol

If you don't mind what is probably cheaper quality, Amazon has some that are under $10.  

Yes, thank you Lazyman, once I knew the correct search term, I managed to find one in brass in middling quality that will work.  Another first for me, never have seen what the mortise looks like.  Appreciate it.