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×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×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).

Not to be confused with Jónsson algebra or Lindenbaum–Tarski 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×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×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.

gollark: I'm not sure how low I could make the time reasonably be without either having to space the frequencies out a lot to keep them distinguishable or making it sound staticy.
gollark: Well, that should be easier, I can just pregenerate... 0.1 second blocks of various frequencies, or something.
gollark: Ah, hmm.
gollark: Also, recognizing frequency would probably be irritating, I'd need a... Fourier transform, or something, or just some really hacky thing for recognizing *one* frequency.
gollark: I see. Well, if you write/find DFPWM to/from... raw signed ints, or something... conversion, I'll look at implementing that.

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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.