Finite Presentability of Brin-Higman-Thompson Monoids via Free J\'onsson-Tarski Algebras

We show that the monoids totM_{k,1} introduced by Birget and their generalizations tot nM_{k,r} which extend the Brin-Higman-Thompson groups, can be realized as the endomorphism monoids of higher-dimensional J\'onsson-Tarski algebras. We also show how elements of these monoids can be thought of as "rewrite rules". We use these representations to show that the monoids are finitely presented.