Decidability of first-order theories for groups and monoids of integral matrices

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.