A Construction of Subshifts and a Class of Semigroups

Subshifts with property $(A)$ are constructed from a class of directed graphs. As special cases the Markov-Dyck shifts are shown to have property $(A)$. The semigroups, that are associated to $\mathcal R$-graph shifts with Property (A), are determined. Also results on the reconstruction of $\mathcal R$-graphs from their $\mathcal R$-graph shifts are obtained.