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We show that the total monoids totnM_{k,1} introduced by Birget(and theirmulti-rooted generalisations), which extend the Brin-Higman-Thompson groups,can be realised as the endomorphism monoids of higher dimensionalJ\'onnson-Tarski algebras. We use this representation to show that they arefinitely presented.

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