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In this paper, we initiate the study of higher rank Baumslag-Solitar semigroups and their related C*-algebras. We focus on two extreme, but interesting, classes - one is related to products of odometers and the other is related to Furstenberg's p, , q conjecture. For the former class, whose C*-algebras are studied by H. Li and the second author, we here characterize the factoriality of the associated von Neumann algebras and further determine their types; for the latter, we obtain their canonical Cartan subalgebras. In the rank 1 case, we study a more general setting which encompasses (single-vertex) generalized Baumslag-Solitar semigroups. One of our main tools is from self-similar higher rank graphs and their C*-algebras.
Valente et al. (Sat,) studied this question.