symmetric monoidal (∞,1)-category of spectra
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 $A$ 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}
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).
self-similarity?
Wikipedia, Jónsson-Tarski algebra
Stackexchange: Jónsson-Tarski algebra form a Schreier variety
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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.