Free Boolean algebra

A Boolean algebra with a system of generators such that every mapping from this system into a Boolean algebra can be extended to a homomorphism. Every Boolean algebra is isomorphic to a quotient algebra of some free Boolean algebra.

For any cardinal number \$\mathfrak{a}\$ there is a unique (up to an isomorphism) free Boolean algebra with \$\mathfrak{a}\$ generators. Its Stone space is the topological product of \$\mathfrak{a}\$ simple colons, that is, it is a dyadic discontinuum.

A finite Boolean algebra is free if and only if its number of elements is of the form \$2^{2^n}\$, where \$n\$ is the number of generators. Such a free Boolean algebra is realized as the algebra of Boolean functions of \$n\$ variables. A countable free Boolean algebra is isomorphic to the algebra of open-closed subsets of the Cantor set. Every set of pairwise-disjoint elements of a free Boolean algebra is finite or countable.

An infinite free Boolean algebra cannot be complete. On the other hand, the cardinality of any infinite complete Boolean algebra is the least upper bound of the cardinalities of its free subalgebras (see [5]).

References

 [1] R. Sikorski, "Boolean algebras" , Springer (1969) [2] D.A. Vladimirov, "Boolesche Algebren" , Akademie Verlag (1978) (Translated from Russian) [3] P.R. Halmos, "Lectures on Boolean algebras" , v. Nostrand (1963) [4] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973) [5] S.V. Kislyakov, "Free subalgebras of complete Boolean algebras, and spaces of continuous functions" Siberian Math. J. , 14 (1973) pp. 395–403 Sibirsk. Mat. Zh. , 14 : 3 (1973) pp. 569–581 DOI 10.1007/BF00967616 corrig. Siberian Math. J. 16 (1975) p.322 DOI 10.1007/BF00967519
How to Cite This Entry:
Free Boolean algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_Boolean_algebra&oldid=37261
This article was adapted from an original article by D.A. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article