nLab
Jónsson-Tarski algebra

Contents

Idea

Loosely speaking, a Jónsson-Tarski algebra is an isomorphism 2 02 0×2 02^{\aleph_0}\cong 2^{\aleph_0}\times 2^{\aleph_0} gone algebra.

Definition

A Jónsson-Tarski algebra, also called a Cantor algebra, is a set AA together with an isomorphism AA×AA\cong A\times A.

More generally, an object AA in a symmetric monoidal category \mathcal{M} together with an isomorphism α:AAA\alpha:A\otimes A\rightarrow A is called a Jónsson-Tarski object, or an idempotent object (Fiore&Leinster 2010).

Properties

  • Clearly (at least in classical mathematics), any Jónsson-Tarski algebra is either empty, a singleton, or infinite.

  • The structure of a Jónsson-Tarski algebra can be described by an algebraic theory, with one binary operation μ\mu and two unary operations λ\lambda and ρ\rho such that μ(λ(x),ρ(x))=x\mu(\lambda(x),\rho(x)) = x, λ(μ(x,y))=x\lambda(\mu(x,y))=x, and ρ(μ(x,y))=y\rho(\mu(x,y))=y.

  • Any two Jónsson-Tarski algebras freely generated from finite non empty sets are isomorphic. It was this property they owe their introduction to (Jónsson&Tarski 1961).

  • The category of Jónsson-Tarski algebras is a topos, the so called Jónsson-Tarski topos 𝒥 2\mathcal{J}_2, and hence is an example for an algebraic variety that is also a topos (cf. Johnstone 1985).

  • The Thompson Group F is the group of order-preserving automorphisms of the free Jónsson-Tarski algebra on one generator.

  • Generalized Jónsson-Tarski algebras of the form XX nX\overset{\simeq}{\to}X^n were considered in group theory by Higman (1974).

References

Revised on July 31, 2014 00:29:42 by Thomas Holder (89.15.236.220)