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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, SpringerVerlag, New York (1964).
Mathematics Subject Classification
06E05 no label found03G05 no label found06B20 no label found03G10 no label found06E20 no label found Forums
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example of Boolean algebras by CWoo
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