# free Boolean algebra

Let $A$ be a Boolean algebra^{} and $X\subseteq A$ such that $\u27e8X\u27e9=A$. In other words, $X$ is a set of generators^{} of $A$. $A$ is said to be *freely generated by $X$*, or that $X$ is a *free set of generators of $A$*, if $\u27e8X\u27e9=A$, and every function $f$ from $X$ to some Boolean algebra $B$ can be extended to a Boolean algebra homomorphism $g$ from $A$ to $B$, as illustrated by the commutative diagram^{} below:

$\text{xymatrix}\mathrm{@}R-=2ptX\text{ar}{[dr]}^{f}\text{ar}{[dd]}_{i}\mathrm{\&}BA\text{ar}{[ur]}_{g}$

where $i:X\to A$ is the inclusion map^{}. By extension^{} of $f$ to $g$ we mean that $g(x)=f(x)$ for every $x\in X$. Any subset $X\subseteq A$ containing $0$ (or $1$) can never be a free generating set for any subalgebra^{} of $A$, for any function $f:X\to B$ such that $f(0)\ne 0$ can never be extended to a Boolean homomorphism.

A Boolean algebra is said to be *free* if it has a free set of generators. If $A$ has $X$ as a free set of generators, $A$ is said to be *free on* $X$. If $A$ and $B$ are both free on $X$, then $A$ and $B$ are isomorphic^{}. This means that free algebras are uniquely determined by its free generating set, up to isomorphisms^{}.

A simple example of a free Boolean algebra is the one freely generated by one element. Let $X$ be a singleton consisting of $a$. Then the set $A=\{0,a,{a}^{\prime},1\}$ is a Boolean algebra, with the obvious Boolean operations identified. Every function from $X$ to a Boolean algebra $B$ singles out an element $b\in B$ corresponding to $a$. Then the function $g:A\to B$ given by $g(a)=b$, $g({a}^{\prime})={b}^{\prime}$, $g(0)=0$, and $g(1)=1$ is clearly Boolean.

The two-element algebra^{} $\{0,1\}$ is also free, its free generating set being $\mathrm{\varnothing}$, the empty set^{}, since the only function on $\mathrm{\varnothing}$ is $\mathrm{\varnothing}$, and thus can be extended to any function.

In general, if $X$ is finite, then the Boolean algebra freely generated by $X$ has cardinality ${2}^{{2}^{|X|}}$, where $|X|$ is the cardinality of $X$. If $X$ is infinite^{}, then the cardinality of the Boolean algebra freely generated by $X$ is $|X|$.

Title | free Boolean algebra |
---|---|

Canonical name | FreeBooleanAlgebra |

Date of creation | 2013-03-22 18:01:23 |

Last modified on | 2013-03-22 18:01:23 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 12 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 06E05 |

Classification | msc 03G05 |

Classification | msc 06B20 |

Classification | msc 03G10 |