What use are groups, monoids, and rings in database computations?
The main answer is that by exploiting semi-group structure, we can build systems that parallelize correctly without knowing the underlying operation (the user is promising associativity).

By using Monoids, we can take advantage of sparsity (we deal with a lot of sparse matrices, where almost all values are a zero in some Monoid).

By using Rings, we can do matrix multiplication over things other than numbers (which on occasion we have done).

The algebird project itself (as well as the issue history) pretty clearly explains what is going on here: we are building a lot of algorithms for aggregation of large data sets, and leveraging the structure of the operations gives us a win on the systems side (which is usually the pain point when trying to productionize algorithms on 1000s of nodes).

Solve the systems problems once for any Semigroup/Monoid/Group/Ring, and then you can plug in any algorithm without having to think about Memcache, Hadoop, Storm, etc...