Isomorphic field of Boolean algebra?

Question

I'm just starting to learn about boolean algebra (i.e. the one used in electric circuits) and I have somewhat of a hard time truly them. I can derive all the basic properties (i.e. DeMorgan's law) but the somewhat strange properties of the multiplicative and additive zeros (i.e. $1 + x = 1$) trip me up.

I'm wondering if there's an isomorphism from Boolean algebra to a possibly somewhat more common and intuitive field so that I can understand Boolean algebra better. For instance, something like $\mathbb Z / \mathbb Z_2$? (Obviously, the two cannot be isomorphic since $\mathbb Z / \mathbb Z_2$ is finite while the Boolean algebra is not, so I am just using it for example purposes).

My algebraic knowledge is limited to basic knowledge of groups,rings,fields and some Galois Theory.

Answer 1

You should not denote the Boolean operation $\vee$ with $+$. For one thing, that suggests that the opeation has inverses, which it doesn't.

In many places on the site, you will find explanations on how the Boolean operations $(\vee,\wedge,\neg)$ can be used to create ring operations $(+, \cdot)$ on the same set. Actually $\wedge$ works for $\cdot$, but you need to define $a+b=(\neg a\wedge b)\vee (a\wedge \neg b)$.

After that, it can be proven that $(R,+,\cdot)$ is isomorphic to a subdirect product of some number of copies of the field of two elements. See Boolean algebra gives rise to a ring. This is related to representing it as a ring of sets of a powerset.