DETERMINATIONS OF WEIGHTED FINITE AUTOMATA OVER COMMUTATIVE IDEMPOTENT MF-SEMIRINGS

The semirings admitting maximal factorizations of any finite dimension are called MF-semirings. We first show that a commutative semiring K is an MF-semiring if and only if K admits a maximal factorization of dimension n >= 2, and if and only if K is a multiplicatively cancellative semiring satisfying the g.c.d. condition. And then, by using above result, we prove that a weighted finite automaton A over a commutative idempotent MF-semiring has a determination if A has the victory property and twins property. Also, some special cases are considered.