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Loosely speaking, a Jónsson-Tarski algebra is an isomorphism gone algebra.
A Jónsson-Tarski algebra, also called a Cantor algebra, is a set together with an isomorphism .
More generally, an object in a symmetric monoidal category together with an isomorphism is called a Jónsson-Tarski object, or an idempotent object (Fiore&Leinster 2010).
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 and two unary operations and such that , , and .
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 1961).
The category of Jónsson-Tarski algebras is a topos, the so called Jónsson-Tarski topos , and hence is an example for an algebraic 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.
Generalized Jónsson-Tarski algebras of the form were considered in group theory by Higman (1974).
Wikipedia, Jónsson-Tarski algebra
B. Jónsson, A. Tarski , On Two Properties of Free Algebras , Math. Scand. 9 (1961) pp.95-101. (pdf)
Marcelo Fiore, Tom Leinster, An abstract characterization of Thompson’s group F , arXiv.math/0508617 (2010). (pdf)
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.
Revised on July 31, 2014 00:29:42
by Thomas Holder