Free Boolean algebra
In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that:
The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John is tall" and "Mary is rich". These generate a Boolean algebra with four atoms, namely:
Other elements of the Boolean algebra are then logical disjunctions of the atoms, such as "John is tall and Mary is not rich, or John is not tall and Mary is rich". In addition there is one more element, FALSE, which can be thought of as the empty disjunction; that is, the disjunction of no atoms.
This example yields a Boolean algebra with 16 elements; in general, for finite n, the free Boolean algebra with n generators has 2natoms, and therefore elements.
If there are infinitely many generators, a similar situation prevails except that now there are no atoms. Each element of the Boolean algebra is a combination of finitely many of the generating propositions, with two such elements deemed identical if they are logically equivalent.
In the language of category theory, free Boolean algebras can be defined simply in terms of an adjunction between the category of sets and functions, Set, and the category of Boolean algebras and Boolean algebra homomorphisms, BA. In fact, this approach generalizes to any algebraic structure definable in the framework of universal algebra.
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