Inverse semigroup shifts over countable alphabets

TL;DR

This paper characterizes shift spaces over countable alphabets that can be structured as inverse semigroups, establishing conditions for their recoding as shift spaces with 1-block operations derived from group operations, and classifying Markovian cases.

Contribution

It provides a characterization of inverse semigroup shift spaces over infinite alphabets and links them to group-based 1-block operations, including a classification of Markovian cases.

Findings

Characterization of inverse semigroup shift spaces over countable alphabets.

Conditions for recoding zero-dimensional inverse semigroups as shift spaces with 1-block operations.

Markovian shift spaces with 1-block inverse semigroup operations are conjugate to a product of a full shift and a fractal shift.

Abstract

In this work we characterize shift spaces over infinite countable alphabets that can be endowed with an inverse semigroup operation. We give sufficient conditions under which zero-dimensional inverse semigroups can be recoded as shift spaces whose correspondent inverse semigroup operation is a 1-block operation, that is, it arises from a group operation on the alphabet. Motivated by this, we go on to study block operations on shift spaces and, in the end, we prove our main theorem, which states that Markovian shift spaces, which can be endowed with a 1-block inverse semigroup operation, are conjugate to the product of a full shift with a fractal shift.