A categorical invariant of flow equivalence of shifts

TL;DR

This paper demonstrates that the Karoubi envelope of a shift's syntactic semigroup is a fundamental invariant under flow equivalence, leading to new classification tools for shifts and related structures.

Contribution

It establishes the Karoubi envelope as the optimal syntactic invariant for flow equivalence of sofic shifts and links it to existing covers and properties.

Findings

Karoubi envelope is a flow invariant for shifts.

Flow equivalence of certain shifts corresponds to graph isomorphism.

Flow invariants include new and classical invariants.

Abstract

We prove that the Karoubi envelope of a shift --- defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift --- is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants --- some new and some old --- are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov-Dyck and Markov-Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property (A) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property (A) is decidable for sofic shifts.