Take for example, topology, one of the 3 “main” branches of abstract, pure math (along with algebra and analysis). Note the definition of a topology: for some set/space X, it contains ∅ and both ∅ and X itself are open, as well as the arbitrary intersection and union both being open.
Note that this means, under union, topologies for monoids: basically groups without an inverse element. Union with empty gives an identity, and it’s very obviously associative; the only thing missing is inverse (this means including the set minus operation \ forms a ring, though). This means that the set of all topologies is a monoid.
Now, let’s look at something on the very far end of math — formal languages. Formal languages form the basis for computer programming languages and are a large traditional tenet of discrete math, a branch of applied math most consider fundamentally different from pure math, about as opposite as you could get from topics like algebra or topology.
Well, formal languages, like topologies, also form monoids under concatenation: a formal language is all the strings over an alphabet of characters with the empty string ε/λ (depending on who’s talking it’ll be epsilon or lambda), and when concatenated with another (put together in a manner such that the strings ‘aba’ and ‘bbaab’ would be ‘ababbaab’) is still in the formal language. The empty string forms the identity and the union of formal languages is associative (and a formal language).
Pretty interesting, huh? Both topologies and formal languages form the same type of abstract structure. Well, since the proof is a little dense, I’m going to outright say it: you can write an isomorphism to show that all topologies with the discrete topology (in which all subsets of the space X are open) can be written as a formal language. Likewise, you can write formal languages as a discrete topology. That is to say, even topology and formal languages are the same structure.
There’s loads of other examples, but this is one of the more opposite ones.