Totally synchronizing generated system

Shahamat, Manouchehr; Ganjbakhsh Sanatee, Ali

doi:10.22080/cjms.2023.25757.1664

Totally synchronizing generated system

Document Type : Research Articles

Authors

¹ ‎Department of Mathematics‎, ‎Dezful branch‎, ‎Islamic Azad University‎, ‎Dezful‎, ‎Iran‎.

² Department of Mathematics, Quchan University of Technology, Quchan, Iran.

10.22080/cjms.2023.25757.1664

Abstract

‎‎We introduce the notion of a minimal generator

G

for the coded system

X

; that is a generator for coded system

X

whenever

u \in G

‎, ‎then

u \notin W (\overset{―}{})

‎. ‎Such an

X

is called \emph{minimally‎ ‎generated system}‎. ‎We ‎aim ‎to‎ introduce a class of minimally‎ ‎generated subshifts generated by some certain synchronizing blocks‎. ‎These systems are precisely the tool that will enable us to show that for such subshifts

X

‎, ‎each

x \in X

‎‎ can be written uniquely as‎

x = \dots v_{- 1} v_{0} v_{1} v_{2} \dots

, where

{\dots ‎, ‎ v_{- 1}, v_{0}, v_{1}, v_{2}, \dots} \in G

.‎‎‎‎‎‎ ‎Shows that the converse of that proposition isn't necessarily true‎.‎ {\color{red}‎ ‎‎‎‎‎‎We will show which of the components of the Kreiger graph of such a subshift could be a candidate to be suitable for a Fischer cover‎.‎‎‎‎‎ }