Relation algebras containing Thompson groups

TL;DR

This paper explores the relationship between various algebraic structures related to relation algebras and Thompson groups, extending representability results to a broader class called J-algebras.

Contribution

It establishes that J-algebras containing conjugated quasiprojections also contain homomorphic images of Thompson groups and extends tabular relation algebra representability to J-algebras.

Findings

J-algebras with conjugated quasiprojections contain Thompson groups

Representability of tabular relation algebras is extended to J-algebras

Connections between relation algebras, Thompson groups, and set theory are clarified

Abstract

The connections between Tarski's relation algebras and Thompson's groups F, T, V, and his monoid M are reviewed here, along with Jonsson-Tarski algebras, fork algebras, true pairing algebras, and tabular relation algebras. All of these algebras are related to the finitization problem and to Tarski's formalization of set theory without variables. Most of the technical details occur in the variety of J-algebras, which is obtained from relation algebras by omitting union and complementation and adopting a set of axioms created by Jonsson. Every relation algebra or J-algebra that contains a pair of conjugated quasiprojections satisfying the Domain and Unicity conditions, such as those that arise from Jonsson-Tarski algebras or fork algebras, will also contain homomorphic images of F, T, V, and M. The representability of tabular relation algebras is extended here to J-algebras, using a notion of tabularity that is equivalent among relation algebras to the original definition.