For an arbitrary finite algebra one defines a double sequence by and .
The problem if such recurrent double sequences are ultimately zero is undecidable, even if we restrict it to the class of commutative finite algebras.
"An Undecidable Property of Recurrent Double Sequences." Notre Dame J. Formal Logic 49 (2) 143 - 151, 2008. https://doi.org/10.1215/00294527-2008-004