On subshift presentations

TL;DR

This paper introduces a new framework for representing subshifts using structured labeled graphs and associated semigroups, establishing conditions for when such representations are possible.

Contribution

It develops a novel method for presenting subshifts via ${ m R}$-graph semigroups and characterizes the necessary and sufficient conditions for these presentations.

Findings

Defines properties (B) and (c) for subshifts

Establishes conditions under which subshifts have ${ m R}$-graph semigroup presentations

Links properties (A), (B), and (c) to topological conjugacy with structured graph 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,