nLab Jónsson-Tarski algebra




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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.


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

More generally, an object AA in a symmetric monoidal category \mathcal{M} together with an isomorphism α:AAA\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 XX together with an isomorphism XX nX\overset{\simeq}{\to}X^n (cf. Smirnov 1971, Higman 1974).1


  • 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 μ\mu and two unary operations λ\lambda and ρ\rho such that μ(λ(x),ρ(x))=x\mu(\lambda(x),\rho(x)) = x, λ(μ(x,y))=x\lambda(\mu(x,y))=x, and ρ(μ(x,y))=y\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 GrpGrp 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 𝒥 2\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).


  • 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.