Why is a finite Boolean algebra mapped to a Powerset, but a linear algebra is not?
I have been studying Discrete mathematics lately, and while studying about Boolean algebra, I wondered what specfici feature about it differs it from other types of algebras?
From my linear algebra class, I remember linear algebra being defined as a vector space with the additional operation of element-multiplication yielding a result within the same vector space.
Boolean algebra can also be defined under the same rules as for linear algebra, we can use XOR for vector addition, with a set of scalars as {0,1} we can also define scalar multiplication and we have the AND operation for vector multiplication.
Would we then also say that this linear algebra is just a powerset in disguise?
I just don't understand how a set which is a collection of elements, with the partially ordered relation of subset, can be equivalent to boolean algebra. It doesn't click for me.