Bratteli diagrams via the De Concini-Procesi theorem

An AF algebra A is said to be an AFl algebra if the Murray-von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AFl algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AFl algebra A, generates a Bratteli diagram of A. We generalize this result to the case when A has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini-Procesi theorem on the elimination of points of indeterminacy in toric varieties.