# nLab Jonsson-Tarski algebra

## Definition

A Jónsson-Tarski algebra is a set $A$ together with an isomorphism $A\cong A×A$.

## 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 $\mu \left(\lambda \left(x\right),\rho \left(x\right)\right)=x$, $\lambda \left(\mu \left(x,y\right)\right)=x$, and $\rho \left(\mu \left(x,y\right)\right)=y$.

• The category of Jónsson-Tarski algebras is a topos, although this is not in general the case for algebraic theories. See here.

Created on December 15, 2010 17:39:57 by Mike Shulman (71.137.3.108)