nLab
Jonsson-Tarski algebra

Definition

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

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

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