Incidentally the typical mathematical curriculum does not even bother with monoids and semigroups when it comes to the initial introduction. The introductory ma

Incidentally the typical mathematical curriculum does not even bother with monoids and semigroups when it comes to the initial introduction. The introductory material jumps right into the definition of a group and then mentions on the side that removing a few key pieces leaves you with a monoid and a semigroup. Same with fields, rings, rigs, etc. The most constrained piece is introduced first and then dropping requirements leaves you with something less constrained for which you can prove fewer theorems. I learned groups and fields first and then learned about the more relaxed versions. I'm not sure which approach is better though. Groups are certainly more fun to play with but monoids probably show up in more contexts.