They were introduced by Bjarni Jónsson and Alfred Tarski (1961, Theorem 5).
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. A Jónsson–Tarski algebra of type 2 is a set A with a product w from A×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.
If w is any bijection from A×A to A then it can be extended to a unique Jónsson–Tarski algebra by letting pi(a) be the projection of w−1(a) onto the ith factor.