Maximal simultaneously nilpotent sets of matrices over antinegative semirings

We study the simultaneously nilpotent index of a simultaneously nilpotent set of matrices over an antinegative commutative semiring S. We find an upper bound for this index and give some characterizations of the simultaneously nilpotent sets when this upper bound is met. In the special case of antinegative semirings with all zero divisors nilpotent, we also find a bound on the simultaneously nilpotent index for all nonmaximal simultaneously nilpotent sets of matrices and establish their cardinalities in case of a finite S.