### 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 .

# model-theory

# set-theory

# boolean-algebra

# algebra

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