This helped me grasp something that I had read from Alexander Stepanov[1] that I hadn't fully understood before (not being familiar with the algebraic terminology):
> I suddenly realized that the ability to add numbers in parallel depends on the fact that addition is associative...In other words, I realized that a parallel reduction algorithm is associated with a semigroup structure type. That is the fundamental point: algorithms are defined on algebraic structures.
I think the use case of building infrastructure for parallel/distributed computation as described above is a nice, concrete example of why using abstract algebra in our programs can be useful. It certainly isn't the only use case though. Other things include managing complex control flow, or passing an implicit context through a computational pipeline.
[0] https://www.infoq.com/presentations/abstract-algebra-analyti...
