Abstract
We deal with the decidability problem for first-order theories of a complete linear group GL(n,ℤ) of all integral matrices of order n ≥ 3. and of a respective complete linear monoid ML(n,ℤ). It is proved that theories ∀¬ ∧ GL(3,ℤ). ∃∀∧ GL(3,ℤ). ∀¬ ∧ ML(3,ℤ), and ∃¬ ∧ ML(3,ℤ) are critical. and that ∃∀ ∧ νGL(n,ℤ) and ∃∀ ∧ML(n,ℤ) are decidable for any n ≥ 3.
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