A Study of 2 x 2 Matrix Semirings over Commutative Weak Inductive *-semirings

Let S = (S, +, ·, *, 0, 1, ≤) be a weak inductive *-semiring. The collection of all n × n matrices S
^n×n
, equipped with the usual matrix operations +, · and the unary operation * defined in [3], form a *-semiring. Esik and Kuich propose a open problem that whether the semiring S
^n×n
is weak inductive. In this paper, we investigate the case n = 2. It is shows that if S is both a commutative semiring and a λ-semiring, then S
^2×2
is weak inductive. By using this result we determine the least simultaneous fixed point of a system of equation proposed in [5], if it is do exist.