$\mathbb{Z}^{2}$-dimension groups

We study a class of simple dimension groups in which the cyclic subgroup generated by the order unit is replaced by a copy of $\mathbb{Z}^{2}$ satisfying some strict conditions. Our main results are necessary and sufficient conditions on a Bratteli diagram which provides inductive limit structures for such groups. This result has an important application in constructing a version of the Bratteli-Vershik model for minimal actions of $\mathbb{Z}^{2}$ on the Cantor set which will be the subject of a subsequent paper.