nLab Jónsson-Tarski algebra

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).

Related concepts

• Jónsson-Tarski topos

• Thompson group

• idempotent

• self-similarity?

Links

• Wikipedia, Jónsson-Tarski algebra

• Stackexchange: Jónsson-Tarski algebra form a Schreier variety

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.

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