On subshift presentations

Abstract: 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}^+$. With additionally given a relation $\mathcal R$ between the edges in ${\mathcal E}^-$ and the edges in $\mathcal E^+ $, and denoting the vertex set of the graph by ${\frak P}$, we speak of an an ${\mathcal R}$-graph ${\mathcal G}{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+) $. From ${\mathcal R}$-graphs ${\mathcal G}{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+) $ we construct semigroups (with zero) ${\mathcal S}{\mathcal R}({\frak P}, {\mathcal E}^-,{\mathcal E}^+) $ that we call ${\mathcal R}$-graph semigroups. We describe a method of presenting subshifts by means of suitably structured labelled directed graphs $({\mathcal V}, \Sigma,\lambda)$ with vertex set ${\mathcal V}$, edge set $\Sigma$, and a label map that asigns to the edges in $\Sigma$ labels in an ${\mathcal R}$-graph semigroup ${\mathcal S}{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$. We call the presented subshift an ${\mathcal S}{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$-presentation. We introduce a Property $(B)$ and a Property (c), 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 ${\mathcal S}{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^-)$, that Properties $(B)$ and (c) 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,