Algebraic set theory

The theory of sets can be described by a boolean algebra equipped with a singleton function which converts between any atom in the boolean algebra into its corresponding member value. Sets can then be described within this algebraic structure entirely in terms of the join operation of the boolean algebra and the singleton function. This allows us to describe sets entirely in terms of algebraic set theory.

The only element in a boolean algebra which does not contain any atoms is the lower bound element of the boolean algebra. By using this lower bound element, the join operation of the boolean algebra, and the singleton function we can produce elements of the pure elements of the algebra. These pure elements correspond to the pure sets which we generally encounter in set theory. Some algebras of sets are limited to only pure elements and others are not.