Franklin's Notes

Boolean algebra

A boolean algebra can be expressed in first-order logic as a model of the language $\mathscr{L} = {+,\cdot,\overline{},0,1}$, where $+$ and $\cdot$ are binary operations (or 2-placed function symbols) and $\overline{}$ is a unary operation (or a 1-placed function symbol), which satisfies the theory consisting of the following axioms:

1. Associativity.
2. Commutativity.
3. Idempotence.
4. Distributivity.
5. Absorption.
6. DeMorgan's laws.
7. Laws of zero and one.
8. Double negation.

A partial order $\leq$ may be defined on a boolean algebra by saying that $x\leq y$ if and only if $x+y=y$. Under this definition, $0$ and $1$ are the smallest and largest elements respectively. Additionally, the least upper bound of two elements $x,y$ is given by $x+y$, while the greatest lower bound is $x\cdot y$.

An element $x$ of a boolean algebra is called an atom if there is no element of $y$ strictly between $0$ and $x$, i.e. $\not\exists y ~ 0<y<x$. The boolean algebra is called atomic if every nonzero element is greater than or equal to ("includes") some atom. On the other hand, it is called atomless if it has no atoms. The additional axiom gives the theory of atomic boolean algebras, whereas the axiom gives the theory of atomless boolean algebras.

Representation theorem for boolean algebras. Every boolean algebra is isomorphic to a field of sets.

The above theorem can be proven by explicitly constructing a field of sets equivalent to the boolean algebra, namely one whose elements are the sets of ultrafilters containing each element of $A$, as demonstrated in this note .