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A Note on the Emptiness of Semigroup Intersections

Bell P.

Fundamenta Informaticae

2007

Vol. 79, nr 1-2

1-4

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.