In this article we survey recent progress in the algorithmic theory of matrix semigroups. The
main objective in this area of study is to construct algorithms that decide various properties …

It is shown that the subgroup membership problem for a virtually free group can be decided
in polynomial time when all group elements are represented by so-called power words, ie …

The main result of this paper is the decidability of the membership problem for 2× 2
nonsingular integer matrices. Namely, we will construct the first algorithm that for any …

We consider the Membership and the Half-Space Reachability problems for matrices in
dimensions two and three. Our first main result is that the Membership Problem is decidable …

We study the identity problem for matrices, ie, whether the identity matrix is in a semigroup
generated by a given set of generators. In particular we consider the identity problem for the …

We consider semigroup algorithmic problems in the Special Affine group
SA(2,Z)=Z^2SL(2,Z), which is the group of affine transformations of the lattice Z^2 that …

We consider semigroup algorithmic problems in finitely generated metabelian groups. Our
paper focuses on three decision problems introduced by Choffrut and Karhum'aki (2005) …

Let G be a unitriangular matrix group of nilpotency class at most ten. We show that the
Identity Problem (does a semigroup contain the identity matrix?) and the Group Problem (is …

We show that the problem of determining if the identity matrix belongs to a finitely generated
semigroup of 2× 2 matrices from the modular group PSL 2 (Z), the Special Linear group SL 2 …

We consider the membership problem for matrix semigroups, which is the problem to decide
whether a matrix belongs to a given finitely generated matrix semigroup. In general, the …