[gue5t]

What structure does an algebra require to merit the name? I've been confused by examples such as the "algebra of a monad", which as I understand it arises from adding an "unwrap" operation in addition to monadic "return" and "join". Most of the references to this only ever refer to specific cases such as an algebra over a monad or an algebra over a functor or a division algebra, etc., and I haven't seen anyone be clear about what makes something an algebra.

[jordigh]

I would expect an algebra to be at least a ring, i.e. an algebra over a field, a vector space with a bilinear product. If you just say plain algebra, I think matrices and vectors or similar.

Most of the structures called algebras (e.g. boolean algebras) have at least two operations and these operations frequently interact via distributivity or something like that. Other examples are ring-like, like a sigma algebra in measure theory.

I've never really encountered anyone saying "algebra" for a generic algebraic structure. I think with this definition Cohn was trying to start a trend that didn't catch on.

[matt-noonan]

There are two different notions of algebra in common usage. One is the definition you gave (vector space with a bilinear product, as in Lie algebras, Clifford algebras, etc). The other definition, as used in the blog, comes from universal algebra. https://en.m.wikipedia.org/wiki/Universal_algebra

Which one(s) you have been exposed to just depends on which mathematical subcultures you’ve interacted with.