Monoids are a key idea in computer science and I think that everyone should know what monoids are. Monoids are EVERYWHERE and once you recognize that you have a monoid you can immediately use known methods to compute over monoids in efficiently (e.g. https://static.googleusercontent.com/media/research.google.c...).
Once you have a certain lattice (I'm leaving out some details), you can solve problems efficiently by known methods (e.g. using dataflow solvers) and automatically have a proof that the solution always terminates.
I really wish that more people took their time to learn these fundamentals and I think this is a wonderful introductory site. I don't mean to offend anyone, but if you've never heard of most of that stuff, then I think that you don't deserve to call yourself a professional programmer. Why? You will reinvent solutions to problems that have been solved before much more efficiently then you will be able to i.e. you're an amateur.