[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.

[jordigh]

Okay, thanks.

Mathematics Wikipedia is sometimes very biased towards certain points of view (usually more undergraddy, as evidenced by its insistence with the "abstract algebra" term), so I think it's being a bit too pushy with its "algebra" definition here. I've edited the term to say that this usage of "algebra" is a universal algebra thing.