free Boolean algebra
Let be a Boolean algebra and such that . In other words, is a set of generators of . is said to be freely generated by , or that is a free set of generators of , if , and every function from to some Boolean algebra can be extended to a Boolean algebra homomorphism from to , as illustrated by the commutative diagram below:
where is the inclusion map. By extension of to we mean that for every . Any subset containing (or ) can never be a free generating set for any subalgebra of , for any function such that 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 has as a free set of generators, is said to be free on . If and are both free on , then and are isomorphic. This means that free algebras are uniquely determined by its free generating set, up to isomorphisms.
A simple example of a free Boolean algebra is the one freely generated by one element. Let be a singleton consisting of . Then the set is a Boolean algebra, with the obvious Boolean operations identified. Every function from to a Boolean algebra singles out an element corresponding to . Then the function given by , , , and is clearly Boolean.
In general, if is finite, then the Boolean algebra freely generated by has cardinality , where is the cardinality of . If is infinite, then the cardinality of the Boolean algebra freely generated by is .
|Title||free Boolean algebra|
|Date of creation||2013-03-22 18:01:23|
|Last modified on||2013-03-22 18:01:23|
|Last modified by||CWoo (3771)|