# nLab Jónsson-Tarski algebra

Contents

### Context

#### Topos Theory

higher algebra

universal algebra

topos theory

# Contents

## Idea

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

## Definition

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

More generally, an object $A$ in a symmetric monoidal category $\mathcal{M}$ together with an isomorphism $\alpha:A\otimes A\rightarrow A$ 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 $\mu$ and two unary operations $\lambda$ and $\rho$ such that $\mu(\lambda(x),\rho(x)) = x$, $\lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$.

• 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 $\mathcal{J}_2$, 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 $X\overset{\simeq}{\to}X^n$ were considered in group theory by Higman (1974).