The k-local Myhill graphs provide an easy means to generalize the suffix substitution closure property for the strictly k-local languages.
Lemma (k-Local Suffix Substitution Closure) If L is a strictly k-local language then for all strings u_{1}, v_{1}, u_{2}, and v_{2} in Σ* and all strings x in Σ^{k-1} :
u_{1}xv_{1} ∈ L and u_{2}xv_{2} ∈ L ⇒ u_{1}xv_{2} ∈ L.
The justi?cation is essentially identical to that of our original suffix substitution closure lemma. If L ∈ SLk then it is recognized by an SLk automaton. In the k-local Myhill graph of that automaton, any path from ‘?' to the vertex labeled x can be put together with any path from that vertex to ‘?' to produce a path that represents a string in L.