A Note on the Emptiness of Semigroup Intersections

We consider decidability questions for the emptiness problem of intersections of matrix semigroups. This problem was studied by A. Markov [7] and more recently by V. Halava and T. Harju [5]. We give slightly strengthened results of their theorems by using a different matrix encoding. In particular, we show that given two finitely generated semigroups of non-singular upper triangular 3 × 3 matrices over the natural numbers, checking the emptiness of their intersections is undecidable. We also show that the problem is undecidable even for unimodular matrices over 3 × 3 rational matrices.

• Type: Journal
• Year: 2007
• Where: FUIN
• Authors: Paul Bell
• Keywords: Different Matrix, Emptiness Problem, FUIN 2007, Matrix Semigroups