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 $x'\in B$ such that

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 $'$ may not be preserved).

Boolean Algebras

A Boolean algebra is a Boolean lattice such that $'$ 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 $'$ . 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 $$x \land y = xy,\qquad x \lor y = x + y + xy,\qquad\mbox{and}\qquad x'=1+x.$$ To view a Boolean lattice as a Boolean ring, define $$xy = x \land y\qquad\mbox{ and }\qquad x + y = (x' \land y) \lor (x \land y').$$

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

Bibliography

1

G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).

2

R. Sikorski, Boolean Algebras, 2nd Edition, Springer-Verlag, New York (1964).