# 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

 $\displaystyle x\land x^{\prime}$ $\displaystyle=0$ $\displaystyle x\lor x^{\prime}$ $\displaystyle=1$ $\displaystyle(x^{\prime})^{\prime}$ $\displaystyle=x$ $\displaystyle(x\land y)^{\prime}$ $\displaystyle=x^{\prime}\lor y^{\prime}$ $\displaystyle(x\lor y)^{\prime}$ $\displaystyle=x^{\prime}\land y^{\prime}$

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

 $x\land y=xy,\qquad x\lor y=x+y+xy,\qquad\mbox{and}\qquad x^{\prime}=1+x.$

To view a Boolean lattice as a Boolean ring, define

 $xy=x\land y\qquad\mbox{ and }\qquad x+y=(x^{\prime}\land y)\lor(x\land y^{% \prime}).$

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

## References

• 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
• 2 R. Sikorski, Boolean Algebras, 2nd Edition, Springer-Verlag, New York (1964).
 Title Boolean lattice Canonical name BooleanLattice Date of creation 2013-03-22 12:27:20 Last modified on 2013-03-22 12:27:20 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 19 Author mathcam (2727) Entry type Definition Classification msc 06E05 Classification msc 03G05 Classification msc 06B20 Classification msc 03G10 Classification msc 06E20 Synonym Boolean algebra Related topic BooleanRing