Semigroup Forum

, Volume 96, Issue 2, pp 203–240

# Inverse semigroup shifts over countable alphabets

• Daniel Gonçalves
• Marcelo Sobottka
• Charles Starling
Research Article

## 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.

## Keywords

Inverse semigroups Symbolic dynamics Shift spaces Markov shifts Topological semigroups Topological dynamics

## Notes

### Acknowledgements

D. Gonçalves was partially supported by CNPq and Capes Project PVE085/2012. M. Sobottka was supported by CNPq-Brazil Grants 304813/2012-5, 480314/2013-6 and 308575/2015-6. Part of this work was carried out while the author was postdoctoral fellow of CAPES-Brazil at Center for Mathematical Modeling, University of Chile. C. Starling was supported by CNPq, and work on this paper occured while the author held a postdoctoral fellowship at UFSC. We thank the referee for many helpful suggestions and for a thorough reading.

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