# nLab Jónsson-Tarski algebra

Contents

### Context

#### Topos Theory

higher algebra

universal algebra

topos theory

# Contents

## Idea

A Jónsson-Tarski algebra is a set that looks like two copies of itself. Since historically the classical examples of these occured in the cardinal arithmetics of Georg Cantor, they are also known as Cantor algebras.

## 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).

In another possible direction for generalization, one defines a Jónsson-Tarski n-algebra as a set $X$ together with an isomorphism $X\overset{\simeq}{\to}X^n$ (cf. Smirnov 1971, Higman 1974).1

## 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 1956,1961).

• Just like in the category $Grp$ of groups, subalgebras of free algebras are free themselves (cf. this Stackexchange question).

• The category of Jónsson-Tarski algebras is a topos, the so called Jónsson-Tarski topos $\mathcal{J}_2$, and hence is an example for a 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 (cf. Fiore-Leinster 2010).

## References

• K. S. Brown, Finiteness Properties of Groups , JPAA 44 (1987) pp.45-75.

• J. Dubeau, Jónsson Jónsson-Tarski algebras , arXiv:2202.02460 (2022). (abstract)

• J. Dudek, A. W. Marczak, On Cantor Identities , Algebra Universalis 68 (2012) pp.237–247.

• Marcelo Fiore, Tom Leinster, An abstract characterization of Thompson’s group F , arXiv.math/0508617 (2010). (pdf)

• R. Freese, J. B. Nation, Free Jónsson-Tarski algebras , ms. 2020. (pdf)

• G. Higman, Finitely presented infinite simple groups , Notes on Pure Mathematics 8 (1974) Australian National University Canberra.

• P. Hines, The Categorical Theory of Self-Similarity , TAC 6 no.3 (1999). (abstract)

• Peter Johnstone, When is a Variety a Topos? , Algebra Universalis 21 (1985) pp.198-212.

• Peter Johnstone, Collapsed Toposes and Cartesian Closed Varieties , JA 129 (1990) pp.446-480.

• B. Jónsson, A. Tarski , Two General Theorems Concerning Free Algebras , Bull. Amer. Math. Soc. 62 p.554. (pdf)

• B. Jónsson, A. Tarski , On Two Properties of Free Algebras , Math. Scand. 9 (1961) pp.95-101. (pdf)

• Tom Leinster, Jónsson-Tarski toposes, Talk Nice 2007. (slides)

• A. K. Rumjancev, An independent basis for the quasi-identities of a free Cantor algebra , Algebra and Logic 16 (1977) pp.119-129.

• D. M. Smirnov, Cantor algebras with a single generator I , Algebra and Logic 10 (1971) pp.40-49.

• D. M. Smirnov, Cantor algebras with a single generator II , Algebra and Logic 12 (1973) pp.399-404.

• D. M. Smirnov, Bases and automorphisms of free Cantor algebras of finite rank , Algebra and Logic 13 (1974) pp.17-33.

• S. Swierczkowski, On isomorphic free algebras , Fund. Math. 50 (1961) pp.35–44.

1. A profunctorial variation on this theme has been proposed by Leinster (2007). See at Jónsson-Tarski topos for some details.

Last revised on May 4, 2022 at 13:03:05. See the history of this page for a list of all contributions to it.