nLab
Jónsson-Tarski algebra

Context

Algebra

Topos Theory

higher algebra

universal algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

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)