We consider partitioned graphs, by which we mean finite directed graphs with a partitioned edge set
${\mathcal{E}}={\mathcal{E}}^{-}\cup {\mathcal{E}}^{+}$
. Additionally given a relation
${\mathcal{R}}$
between the edges in
${\mathcal{E}}^{-}$
and the edges in
${\mathcal{E}}^{+}$
, and under the appropriate assumptions on
${\mathcal{E}}^{-},{\mathcal{E}}^{+}$
and
${\mathcal{R}}$
, denoting the vertex set of the graph by
$\mathfrak{P}$
, we speak of an
${\mathcal{R}}$
-graph
${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$
. From
${\mathcal{R}}$
-graphs
${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$
we construct semigroups (with zero)
${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$
that we call
${\mathcal{R}}$
-graph semigroups. We write a list of conditions on a topologically transitive subshift with property
$(A)$
that together are sufficient for the subshift to have an
${\mathcal{R}}$
-graph semigroup as its associated semigroup.
Generalizing previous constructions, we describe a method of presenting subshifts by means of suitably structured finite labeled directed graphs
$({\mathcal{V}},~\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706}~)$
with vertex set
${\mathcal{V}}$
, edge set
$\unicode[STIX]{x1D6F4}$
, and a label map that assigns to the edges in
$\unicode[STIX]{x1D6F4}$
labels in an
${\mathcal{R}}$
-graph semigroup
${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$
. We denote the presented subshift by
$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$
and call
$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$
an
${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$
-presentation.
We introduce a property
$(B)$
of subshifts that describes a relationship between contexts of admissible words of a subshift, and we introduce a property
$(c)$
of subshifts that in addition describes a relationship between the past and future contexts and the context of admissible words of a subshift. Property
$(B)$
and the simultaneous occurrence of properties
$(B)$
and
$(c)$
are invariants of topological conjugacy.
We consider subshifts in which every admissible word has a future context that is compatible with its entire past context. Such subshifts we call right instantaneous. We introduce a property
$RI$
of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a right instantaneous presentation. We consider also subshifts in which every admissible word has a future context that is compatible with its entire past context, and also a past context that is compatible with its entire future context. Such subshifts we call bi-instantaneous. We introduce a property
$BI$
of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a bi-instantaneous presentation.
We define a subshift as strongly bi-instantaneous if it has for every sufficiently long admissible word
$a$
an admissible word
$c$
, that is contained in both the future context of
$a$
and the past context of
$a$
, and that is such that the word
$ca$
is a word in the future context of
$a$
that is compatible with the entire past context of
$a$
, and the word
$ac$
is a word in the past context of
$a$
, that is compatible with the entire future context of
$a$
. We show that a topologically transitive subshift with property
$(A)$
, and associated semigroup a graph inverse semigroup
${\mathcal{S}}$
, has an
${\mathcal{S}}$
-presentation, if and only if it has properties
$(c)$
and
$BI$
, and a strongly bi-instantaneous presentation, if and only if it has properties
$(c)$
and
$BI$
, and all of its bi-instantaneous presentations are strongly bi-instantaneous.
We construct a class of subshifts with property
$(A)$
, to which certain graph inverse semigroups
${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$
are associated, that do not have
${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$
-presentations.
We associate to the labeled directed graphs
$({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$
topological Markov chains and Markov codes, and we derive an expression for the zeta function of
$X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$
in terms of the zeta functions of the topological Markov shifts and the generating functions of the Markov codes.