Some decision problems on integer matrices
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 125-131.

Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free; 2) contains the identity matrix; 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs to a given finitely generated semigroup.

DOI : 10.1051/ita:2005007
Classification : 20M05, 68R15 
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     title = {Some decision problems on integer matrices},
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Choffrut, Christian; Karhumäki, Juhani. Some decision problems on integer matrices. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 125-131. doi : 10.1051/ita:2005007. https://numdam.org/articles/10.1051/ita:2005007/

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