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 $2r={(2r)}^{2}=4{r}^{2}=4r$,

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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.
References
 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
 2 R. Sikorski, Boolean Algebras, 2nd Edition, SpringerVerlag, New York (1964).
Title  Boolean lattice 
Canonical name  BooleanLattice 
Date of creation  20130322 12:27:20 
Last modified on  20130322 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 