To associate a topological dynamical system to an ordered Bratteli diagram usually one assumes that the ordering is '{\it proper}'. Then the model for the dynamical system is the infinite path space and the Vershik shift. If the ordering is non-proper, the Vershik shift is still defined on the non-maximal paths. It is a delicate question whether this extends continuously to the maximal paths. Usually it doesn't. We show how to associate a dynamical system to a non-properly ordered Bratteli diagram and describe the $K$-group $K_0$ of the dynamical system in terms of the Bratteli diagram. In the case of properly ordered Bratteli diagrams this description coincides with what is already known, namely the so-called dimension group of the Bratteli diagram. The new ordered group we will define is more relevant for non-properly ordered Bratteli diagrams. This gives a way to compute the $K_0$ of a (non-proper) substitutional system, which, so far has had an elegant description only for properly ordered substitutions.