free Boolean algebra

Let A be a Boolean algebraMathworldPlanetmath and XA such that X=A. In other words, X is a set of generatorsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of A. A is said to be freely generated by X, or that X is a free set of generators of A, if X=A, and every function f from X to some Boolean algebra B can be extended to a Boolean algebra homomorphism g from A to B, as illustrated by the commutative diagramMathworldPlanetmath below:


where i:XA is the inclusion mapMathworldPlanetmath. By extensionPlanetmathPlanetmathPlanetmath of f to g we mean that g(x)=f(x) for every xX. Any subset XA containing 0 (or 1) can never be a free generating set for any subalgebraPlanetmathPlanetmathPlanetmath of A, for any function f:XB such that f(0)0 can never be extended to a Boolean homomorphism.

A Boolean algebra is said to be free if it has a free set of generators. If A has X as a free set of generators, A is said to be free on X. If A and B are both free on X, then A and B are isomorphicPlanetmathPlanetmathPlanetmath. This means that free algebras are uniquely determined by its free generating set, up to isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

A simple example of a free Boolean algebra is the one freely generated by one element. Let X be a singleton consisting of a. Then the set A={0,a,a,1} is a Boolean algebra, with the obvious Boolean operations identified. Every function from X to a Boolean algebra B singles out an element bB corresponding to a. Then the function g:AB given by g(a)=b, g(a)=b, g(0)=0, and g(1)=1 is clearly Boolean.

The two-element algebraMathworldPlanetmath {0,1} is also free, its free generating set being , the empty setMathworldPlanetmath, since the only function on is , and thus can be extended to any function.

In general, if X is finite, then the Boolean algebra freely generated by X has cardinality 22|X|, where |X| is the cardinality of X. If X is infiniteMathworldPlanetmath, then the cardinality of the Boolean algebra freely generated by X is |X|.

Title free Boolean algebra
Canonical name FreeBooleanAlgebra
Date of creation 2013-03-22 18:01:23
Last modified on 2013-03-22 18:01:23
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 06E05
Classification msc 03G05
Classification msc 06B20
Classification msc 03G10