On subshift presentations

We consider partitioned graphs, by which we mean finite strongly connected directed graphs with a partitioned edge set $ {\mathcal E} ={\mathcal E}^- \cup{\mathcal E}^+$\mathrm{.}Withadditionallygivenarelation$\mathcal R$ between the edges in ${\mathcal{E}}^{-}$ and the edges in ${\mathcal{E}}^{+}$, and denoting the vertex set of the graph by $\mathfrak{P}$, we speak of an an ${\mathcal R}$-graph${\mathcal G}{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+) $\mathrm{.}From${\mathcal R}$-graphs${\mathcal G}{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+) $weconstructsemigroups(withzero)${\mathcal S}{\mathcal R}({\frak P}, {\mathcal E}^-,{\mathcal E}^+) $thatwecall${\mathcal R}$-graph semigroups. We describe a method of presenting subshifts by means of suitably structured labelled directed graphs $({\mathcal V}, \Sigma,\lambda)$withvertexset${\mathcal V}$,edgeset$\Sigma$, and a label map that asigns to the edges in $\mathrm{\Sigma }$ labels in an $\mathcal{R}$-graph semigroup ${\mathcal S}{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$\mathrm{.}Wecallthepresentedsubshiftan${\mathcal S}{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$-presentation. We introduce a Property $(B)$ and a Property ©, tof subshifts, and we introduce a notion of strong instantaneity. Under an assumption on the structure of the $\mathcal{R}$-graphs ${\mathcal G}{\mathcal R}({\frak P},{\mathcal E}^-, {\mathcal E}^-)$ we show for strongly instantaneous subshifts with Property $(A)$ and associated semigroup ${\mathcal S}{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^-)$,thatProperties$(B)$ and © are necessary and sufficient for the existence of an ${\mathcal S}{\mathcal R}({\frak P}, {\mathcal E}^-,{\mathcal E}^-)$-presentation, to which the subshift is topologically conjugate,