6
$\begingroup$

During my studies I have come across two different notions of the term "algebra", namely algebra over a set and algebra over a field (the field its vector space always being Euclidean space in my case).

While both of those concepts use the same term, I can't seem to find any relation between them, e.g. one being a sub-case of the other. Is there any such relation? Is there a reason why they are called the same?

$\endgroup$
0

1 Answer 1

6
$\begingroup$

There is a connection.

An algebra over a set is actually a boolean ring: if we interpret $+$ as symmetric difference $\triangle$ and $\cdot$ as intersection $\cap$, they satisfy all (unital) ring axioms (with $0=\emptyset$ and $1$ equal to the universe), and moreover $x+x=x\triangle x=\emptyset=0$. (However, it is never a field, unless it has two elements, despite being called a field of sets: no element except $1$ itself has a $\cap$-inverse.)

Every boolean ring is trivially an algebra over the two-element field, so there (moreover, every ring is an algebra over the ring of integers, but that is not a field).

But this is, I think, mostly incidental. As far as I can tell, historically, the term algebra was used in a sense closer to the one used in universal algebra now, which is rather more abstract than modern "algebra over a field".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.