A finite subset of a Boolean algebra, generates a finite subalgebra.

Question

Let $\mathscr{B}$ be a Boolean algebra and let $F$ be a finite subset of $\mathscr{B}$. Is $\left<F\right>$, the subalgebra generated by $F$, necessarily a finite subalgebra of $\mathscr{B}$?

I thinks that this is true based on following argument: Denote the elements of $F$ by $x_1,\ldots,x_n$, and consider the $2^n$ possible ways of forming $\land$(meet) of $n$ elements $x_{1}^{e_1},\ldots,x_{i}^{e_i},\ldots x_{n}^{e_n}$ where each $e_i$ can be either the empty word, or the complement symbol '. Let $A$ be the subset of $\mathscr{B}$ having just those $2^n$ values. Some of these $2^n$ ways might yield same elements of $\mathscr{B}$, hence we have a subset $A$ of $\mathscr{B}$ with at most $2^n$ elements. Then $\left<F\right>$ is isomorphic to $\mathscr{P}(A)$, the Boolean algebra of subsets of a set having $|A|$ elements.

I came up with this idea while pondering upon the image of a venn diagram with $n$ circles. I think that this argument is valid, and can be completed in a rigorous way, but I am not sure. Is this a valid argument? Is the statement true?

Answer 1

The statement is right, but your proof is not correct. For example, suppose $F=\{x_1, x_2\}$ with $x_1< x_2$ (that is, $x_1\not=x_2$ and $x_1\vee x_2=x_2$).

Then the $A$ we get consists of $$a_1=x_1\wedge x_2,\quad a_2=x_1\wedge x_2', \quad a_3=x_1'\wedge x_2,\quad a_4=x_1'\wedge x_2'.$$ But distinct elements of the powerset of $A$ do not necessarily correspond to distinct elements of $\langle F\rangle$! Namely, both $\emptyset$ and $\{a_3\}$ correspond to $\bot$, since $x_1<x_2$.

So really, the right statement is that $\langle F\rangle$ is a homomorphic image, or quotient, of $\mathcal{P}(A)$.

Answer 2

For finite $m$ and $n,$ an $n$-generated Boolean algebra has at most $2^n$ atoms, and a finite Boolean algebra with $m$ atoms has exactly $2^m$ elements. A free Boolean algebra on $n$ generators has exactly $2^n$ atoms (which you described) and exactly $2^{2^n}$ elements.

Answer 3

The free Boolean ring on $n$ generators is nothing more than the polynomial ring $\mathbb{F}_2[x_1,x_2,\dots,x_n]$ modulo the ideal generated by $x_i^2-x_i$, for $i=1,2,\dots n$, so it's essentially the set of linear combinations of monomials having degree at most $1$ in each variable, with coefficients in $\mathbb{F}_2$; there are $2^n$ such monomials, so the free Boolean ring on $n$ generators has $2^{2^n}$ elements. Since Boolean rings and Boolean algebras are the same thing, the free Boolean algebra on $n$ generators is finite.

The subalgebra generated by $F$ is a homomorphic image of a free algebra on $|F|$ generators, so it is finite as well.

Note that the homomorphism need not be injective; for instance, the Boolean subalgebra of $A=\mathscr{P}(\{1,2\})$ generated by $\{1\}$ and $\{2\}$ is $A$ itself, but the free algebra on two elements has $16$ elements. The fact is that between elements of $F$ there can be nontrivial relations, in this case $\{2\}$ is the complement of $\{1\}$.