# Isomorphic field of Boolean algebra?

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.

• Take an algebra of sets maybe with symmetric difference and intersection
– user515599
Jan 18, 2020 at 9:00

## 1 Answer

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.