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

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?

## 3 Answers

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 $\mathcal{P}(A)$.

• But isn't it true that $\left<F\right>$ is isomorphic to $\mathscr{P}(A)$?. I believe your point is that the $A$ I constructed has no reason to be a subalgebra. But I guess they are the atoms of $\left<F\right>$ so that $\left<F\right>$ is a finite Boolean algebra with $|A|$ atoms... thus isomorphic to $\mathscr{P}(A). – Dilemian Dec 22 '16 at 10:51 • @Dilemian I misread, and have edited my answer. There's still a problem, though:$P(A)$might be bigger than$\langle F\rangle$. – Noah Schweber Dec 22 '16 at 20:21 The free Boolean ring on$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$\{1\}$. • That's a neat argument. I never realized that the perspective of Boolean rings made free Boolean algebras so simple to describe. – Eric Wofsey Dec 22 '16 at 21:16 • @EricWofsey I always get lost with prenex form and the like, so I just switch to rings which I'm more familiar with. – egreg Dec 22 '16 at 21:28 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.