Boolean lattice (original) (raw)

Boolean Lattices

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∈R, r2=r. It is easy to see that

• •
  any Boolean ring has characteristic 2, for 2⁢r=(2⁢r)2=4⁢r2=4⁢r,
• •
  and hence a commutative ring, for a+b=(a+b)2=a2+a⁢b+b⁢a+b2=a+a⁢b+b⁢a+b, so 0=a⁢b+b⁢a, and therefore a⁢b=a⁢b+a⁢b+b⁢a=b⁢a.

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).