  Semigroup C*-algebras for right-angled Artin monoids were introduced and
studied by Crisp and Laca. In the paper at hand, we are able to present the
complete answer to their question of when such C*-algebras are isomorphic. The
answer to this question is presented both in terms of properties of the graph
defining the Artin monoids as well as in terms of classification by K-theory,
and is obtained using recent results from classification of non-simple
C*-algebras.
  Moreover, we are able to answer another natural question: Which of these
semigroup C*-algebras for right-angled Artin monoids are isomorphic to graph
algebras? We give a complete answer, and note the consequence that many of the
C*-algebras under study are semiprojective.
