Factorizable embeddings and the period of an irreducible sofic shift

TL;DR

This paper characterizes when subshifts can embed into irreducible sofic shifts via various types of factor maps, extending previous results by introducing concepts like the period of an irreducible sofic shift.

Contribution

It provides necessary and sufficient conditions for embeddings of subshifts into irreducible sofic shifts through different types of factor codes, generalizing MacDonald's results.

Findings

Conditions for embedding subshifts into irreducible sofic shifts.

Introduction of the concept of the period of an irreducible sofic shift.

Multiple equivalent formulations of the period.

Abstract

Generalizing a result of MacDonald we give necessary and sufficient conditions for an arbitrary subshift to embed into an irreducible sofic shift factoring through a given cover by an irreducible subshift of finite type (SFT). We obtain also necessary and sufficient conditions for an arbitrary subshift to embed into an irreducible sofic shift factoring through \emph{some} sliding block code out of an irreducible SFT. We do that when the code is required to be surjective, and hence a factor code, and when it is required to be injective or almost invertible, or is allowed to be arbitrary. These results require concepts of the period of an irreducible sofic shift as well as a concept of a $p$-periodic subshift. Several equivalent formulations of the period are developed.