Jónsson–Tarski algebra

In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set $X$ onto the product $X \times X$ . They were introduced by Bjarni Jónsson 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 $X \times X$ . 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 Jónsson–Tarski algebra on one generator is the Thompson group $F$ .

Definition

A Jónsson–Tarski algebra of type 2 is a set $A$ with a product $w$ from $A \times A$ to $A$ and two 'projection' maps $p 1$ and $p 2$ from $A$ to $A$ , satisfying $p 1 (w (a 1, a 2)) = a 1$ , $p 2 (w (a 1, a 2)) = a 2$ , and $w (p 1 (a), p 2 (a)) = a$ . The definition for type > 2 is similar but with $n$ projection operators.

Example

If $w$ is any bijection from $A \times A$ to $A$ then it can be extended to a unique Jónsson–Tarski algebra by letting $p i (a)$ be the projection of $w -1 (a)$ onto the $i$ th factor.

References

Jónsson, Bjarni; Tarski, Alfred (1961), "On two properties of free algebras", Math. Scand., 9: 95–101, MR 0126399, Zbl 0111.02002
Smirnov, D. M. (1971), "Cantor algebras with one generator. I.", Algebra and Logic, 10: 40–49, doi:10.1007/BF02217801, MR 0296006, Zbl 0223.08006