In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Jónsson–Tarski algebra or Cantor algebra is an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
encoding a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
from an
infinite set onto the
product . They were introduced by . , named them after
Georg Cantor because of Cantor's
pairing function
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.
Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural ...
and Cantor's theorem that an infinite set has the same number of elements as . The term ''Cantor algebra'' is also occasionally used to mean the
Boolean algebra of all
clopen subset
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
s 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 ...
, or the 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 sets (sometimes called the
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 ...
).
The group of
order-preserving automorphisms of the
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
Jónsson–Tarski algebra on one
generator
Generator may refer to:
* Signal generator, electronic devices that generate repeating or non-repeating electronic signals
* Electric generator, a device that converts mechanical energy to electrical energy.
* Generator (circuit theory), an eleme ...
is the
Thompson group .
Definition
A Jónsson–Tarski algebra of type 2 is a set with a product from to and two 'projection' maps and from to , satisfying , , and . The definition for type > 2 is similar but with projection operators.
Example
If is any bijection from to then it can be extended to a unique Jónsson–Tarski algebra by letting be the projection of onto the th factor.
References
*
*
{{DEFAULTSORT:Jonsson-Tarski algebra
Algebraic structures