In mathematics, a Cantor algebra, named after
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
, is one of two closely related
Boolean algebras
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a gen ...
, one
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
and one
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
.
The countable Cantor algebra is the Boolean algebra of all
clopen
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open set, open and closed set, closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but ...
subsets of the
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Thr ...
. This is the
free Boolean algebra In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called ''generators'', such that:
#Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean opera ...
on a countable number of generators. Up to isomorphism, this is the only nontrivial Boolean algebra that is both countable and atomless.
The complete Cantor algebra is the complete Boolean algebra of
Borel subset
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named a ...
s of the reals modulo
meager set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...
s . It is isomorphic to the completion of the countable Cantor algebra. (The complete Cantor algebra is sometimes called the Cohen algebra, though "
Cohen algebra
In mathematical set theory, a Cohen algebra, named after Paul Cohen, is a type of Boolean algebra used in the theory of forcing. A Cohen algebra is a Boolean algebra whose completion is isomorphic to the completion of a free Boolean algebra I ...
" usually refers to a different type of Boolean algebra.) The complete Cantor algebra was studied by von Neumann in 1935 (later published as ), who showed that it is not isomorphic to the
random algebra
In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra ...
of Borel subsets modulo measure zero sets.
References
*
*{{Citation , last1=von Neumann , first1=John , author1-link=John von Neumann , title=Continuous geometry , origyear=1960 , url=https://books.google.com/books?id=onE5HncE-HgC , publisher=
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial su ...
, series=Princeton Landmarks in Mathematics , isbn=978-0-691-05893-1 , mr=0120174 , year=1998
Forcing (mathematics)
Boolean algebra