In mathematics, a J贸nsson鈥揟arski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X脳X. They were introduced by Bjarni J贸nsson and Alfred Tarski ( 1961, Theorem 5). Smirnov ( 1971), named them after Georg Cantor because of Cantor's pairing function and Cantor's theorem that an infinite set X has the same number of elements as X脳X. The term Cantor algebra is also occasionally used to mean the Boolean algebra of all clopen subsets of the Cantor set, or the Boolean algebra of Borel subsets of the reals modulo meager sets (sometimes called the Cohen algebra).
The group of order-preserving automorphisms of the free J贸nsson鈥揟arski algebra on one generator is the Thompson group F.
Definition
A J贸nsson鈥揟arski algebra of type 2 is a set A with a product w from A脳A to A and two 'projection' maps p1 and p2 from A to A, satisfying p1(w(a1,a2)) = a1, p2(w(a1,a2)) = a2, and w(p1(a),p2(a)) = a. The definition for type > 2 is similar but with n projection operators.
Example
If w is any bijection from A脳A to A then it can be extended to a unique J贸nsson鈥揟arski algebra by letting pi(a) be the projection of w鈭1(a) onto the ith factor.
References
- J贸nsson, Bjarni; Tarski, Alfred (1961), "On two properties of free algebras", Math. Scand., 9: 95鈥101, MR 0126399, Zbl 0111.02002
- Smirnov, D. M. (1971), "Cantor algebras with one generator. I.", Algebra and Logic, 10: 40鈥49, doi: 10.1007/BF02217801, MR 0296006, Zbl 0223.08006
In mathematics, a J贸nsson鈥揟arski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set X onto the product X脳X. They were introduced by Bjarni J贸nsson and Alfred Tarski ( 1961, Theorem 5). Smirnov ( 1971), named them after Georg Cantor because of Cantor's pairing function and Cantor's theorem that an infinite set X has the same number of elements as X脳X. The term Cantor algebra is also occasionally used to mean the Boolean algebra of all clopen subsets of the Cantor set, or the Boolean algebra of Borel subsets of the reals modulo meager sets (sometimes called the Cohen algebra).
The group of order-preserving automorphisms of the free J贸nsson鈥揟arski algebra on one generator is the Thompson group F.
Definition
A J贸nsson鈥揟arski algebra of type 2 is a set A with a product w from A脳A to A and two 'projection' maps p1 and p2 from A to A, satisfying p1(w(a1,a2)) = a1, p2(w(a1,a2)) = a2, and w(p1(a),p2(a)) = a. The definition for type > 2 is similar but with n projection operators.
Example
If w is any bijection from A脳A to A then it can be extended to a unique J贸nsson鈥揟arski algebra by letting pi(a) be the projection of w鈭1(a) onto the ith factor.
References
- J贸nsson, Bjarni; Tarski, Alfred (1961), "On two properties of free algebras", Math. Scand., 9: 95鈥101, MR 0126399, Zbl 0111.02002
- Smirnov, D. M. (1971), "Cantor algebras with one generator. I.", Algebra and Logic, 10: 40鈥49, doi: 10.1007/BF02217801, MR 0296006, Zbl 0223.08006