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Side note: the study of monoids isn't a very big field in mathematics, because they're too simple to say too much about them. But if you modify them just a bit, either in the direction of groups or in the direction of categories, you get massive fields of study.

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I'm reminded of John Baez writing recently (https://twitter.com/johncarlosbaez/status/124647011232195379...) about commutative semigroups. Even simpler, but still lots to say!

I probably still stand corrected, but commutativity is a much stronger condition than the existence of an identity!

I think the existence of an identity isn't very important. You can adjoin an identity to any semigroup to turn it into a monoid.

In any case, doesn't adding more properties make a theory simpler? For example the classification of finite simple groups is much harder than the classification of finite simple commutative groups.

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