The Syntactic Graph of a Sofic Shift

Abstract

We define a new invariant for shift equivalence of sofic shifts. This invariant, that we call the syntactic graph of a sofic shift, is the directed acyclic graph of characteristic groups of the non null regular D-classes of the syntactic semigroup of the shift.

A Hierarchy of Irreducible Sofic Shifts

“…Thus these two shifts are not conjugate since their degrees are different. Remark that they have the same syntactic graph, which is another conjugacy invariant defined and described in [2].…”

“…In this section, we recall the definition and the structure of the syntactic semigroup of an irreducible sofic shift [2].…”

“…Before proving Proposition 2, we recall some results from [2] about the syntactic semigroup of a bipartite shift. Let X (respectively Y ) be an irreducible sofic shift whose symbolic adjacency matrices of its right Fischer cover is a (Q × Q)matrix (respectively (Q × Q )-matrix) denoted by A (respectively by B).…”

“…A bipartite shift is associated in a natural way to a pair of elementary strong shift equivalent and irreducible sofic shifts [16]. Another syntactic conjugacy invariant called the syntactic graph of the sofic shift was defined in [2]. We give an example of two non-almost finite type shifts with different degrees (and therefore not conjugate) that have the same syntactic graph.…”