For groups, there is a solid foundation to classify them https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
What art exists for monoids?
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Work has been done, but monoids and semigroups are MUCH more complicated than groups. See here and here for some references. There are lots of deep connections to formal language theory and automata, so you might also want to look in those kinds of references to get a taste of how complicated things can be. – HallaSurvivor May 02 '21 at 19:51
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Thank you! Partially. – xvorsx May 02 '21 at 21:29
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I guess to start, how about commutative semigroups? – xvorsx May 02 '21 at 21:31