Bratteli graphs

We introduce a purely graph-theoretic model for separable approximately finite-dimensional (AF) C*-algebras, called a Bratteli graph, which unlike Bratteli diagrams does not record vertex levels and multiplicities. The graph inverse semigroup and its tight groupoid recover the AF C*-algebra and its Str\u{a}til\u{a}--Voiculescu diagonal. We identify the primitive ideal space with a quotient of the maximal-path space by the relation of mutual cofinality, and use this description to give graph criteria for various C*-algebra properties, and a graph-theoretic correspondence for the ideal lattice.