I’ve taken a look at a number of introductory books on abstract algebra. They all treat groups, rings, and fields, and many of them treat galois theory, linear algebra, algebras over fields.

But none of them treat monoids as a general class (only groups as a special case). Why is this? Why are monoids not considered an essential part of an algebra course, given that they are very general and appear often without inverses, and basically underlie category theory?

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    $\begingroup$ Since every group is a special monoid, they are treated. But the point is that groups appear more often and provide more structure. $\endgroup$ – James Mar 8 at 8:51
  • $\begingroup$ Groups are easier to work with when they come alone, to deal with monoids or semigroups (which appear very often in nature as well) you often need other kinds of structure, like a topology for instance, so they can hardly appear in an abstract algebra course $\endgroup$ – Max Mar 8 at 8:58
  • $\begingroup$ @Max, if that is indeed the explanation (that you need other kind of structure to work with them), then I’d really like to get an illustration of that. I.e. an illustration of a case where having this extra structure makes them workable but omitting the structure would make them non workable. $\endgroup$ – user56834 Mar 8 at 9:02
  • $\begingroup$ I don't know enough about this to know that it's the explanation (though it goes hand in hand with Qiaochu's answer), but an example is semigroups vs right-continuous compact semigroups : there is a well developped theory of the latter (relating for instance to dynamical systems) whereas bare semigroups are hardly workable. For instance proving that $\gamma\mathbb{N}$ (semigroup of nonprincipal ultrafilters) has an idempotent is best done with topological considerations, and this gives a remarkable proof of Hindman's theorem) $\endgroup$ – Max Mar 8 at 9:46
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    $\begingroup$ I don't think most category theorists would describe monoids as "underlying category theory". A category is a horizontal categorification of monoids, but this view is usually not that insightful. Viewing a category as a vertical categorification of partially ordered sets is usually much more insightful and even then most category theorists would not say that order theory "underlies" category theory, though it would be a much more defensible position. $\endgroup$ – Derek Elkins Mar 8 at 10:20

Every mathematical definition needs to navigate a tradeoff between generality and power. Monoids are more general than groups, but the price you pay for that generality is that it's much harder to say things about monoids than it is to say things about groups. Groups have the isomorphism theorems, Lagrange's theorem, the Sylow theorems, etc. etc.; all good stuff with lots of applications. There are much fewer useful general statements like this that you can make about monoids.

In fact I don't know a single useful general theorem about monoids off the top of my head. Probably they exist but I haven't needed to learn them; I think I've seen Benjamin Steinberg quote a few.

If you want to have some fun you can try working out what the correct substitute for a normal subgroup is for monoids, for the purposes of constructing quotients (hint: it is not a kind of submonoid). This is a nice exercise but also I have never used the fact that I know how to do this to prove anything.

  • $\begingroup$ I am not sure of what you are referring to in your last paragraph. Could you elaborate? $\endgroup$ – J.-E. Pin Mar 13 at 10:50

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