Boolean lattice

In this entry, the notions of a Boolean lattice, a Boolean algebra, and a Boolean ring are defined, compared and contrasted.

Boolean Lattices

A Boolean lattice $B$ is a distributive lattice in which for each element $x\in B$ there exists a complement $x^{\prime }\in B$ such that

$x\wedge x^{\prime }$$=0$

$x\vee x^{\prime }$$=1$

${(x^{\prime })}^{\prime }$$=x$

${(x\wedge y)}^{\prime }$$=x^{\prime }\vee y^{\prime }$

${(x\vee y)}^{\prime }$$=x^{\prime }\wedge y^{\prime }$

In other words, a Boolean lattice is the same as a complemented distributive lattice. A morphism between two Boolean lattices is just a lattice homomorphism (so that $0,1$ and ${}^{\prime }$ may not be preserved).

Boolean Algebras

A Boolean algebra is a Boolean lattice such that ${}^{\prime }$ and $0$ are considered as operators (unary and nullary respectively) on the algebraic system. In other words, a morphism (or a Boolean algebra homomorphism) between two Boolean algebras must preserve $0,1$ and ${}^{\prime }$. As a result, the category of Boolean algebras and the category of Boolean lattices are not the same (and the former is a subcategory of the latter).

Boolean Rings

A Boolean ring is an (associative) unital ring $R$ such that for any $r\in R$, $r^2=r$. It is easy to see that

• any Boolean ring has characteristic $2$, for $2r={(2r)}^2=4r^2=4r$,

• and hence a commutative ring, for $a+b={(a+b)}^2=a^2+ab+ba+b^2=a+ab+ba+b$, so $0=ab+ba$, and therefore $ab=ab+ab+ba=ba$.

Boolean rings (with identity, but allowing 0=1) are equivalent to Boolean lattices. To view a Boolean ring as a Boolean lattice, define

$$x\wedge y=xy,x\vee y=x+y+xy,\text{and}\mathit{\hspace{1em}\hspace{1em}}{x}^{\prime }=1+x.$$

To view a Boolean lattice as a Boolean ring, define

$$xy=x\wedge y\mathit{\hspace{1em}\hspace{1em}}\text{ and }\mathit{\hspace{1em}\hspace{1em}}x+y=(x^{\prime }\wedge y)\vee (x\wedge y^{\prime }).$$

The category of Boolean algebras is naturally equivalent to the category of Boolean rings.