The equality (zeroness) problem is decidable for recognisable series over a finite alphabet (paper §4). The decider $decZero$ reads, as a $Bool$, the linear-algebra condition that the initial vector annihilates the subspace reachable from the final vector: $r.sem = 0$ iff that annihilation holds ($semZero_iff_annihilatesReachable$), and the annihilation condition is decidable (it lives in the finite-dimensional space $Fin r.dim → ℚ$), so $r.sem = 0$ is.