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We define right and left invariant matrices as Boolean matrices that are solutions to certain systems of matrix equations and inequalities over additively idempotent semirings. We provide improved algorithms for computing the greatest right and left invariant equivalence and quasi-order matrices. The improvements are based on the usage of the well-known partition refinement technique. Afterwards, we present the application of right invariant matrices in the determinization of weighted automata over additively idempotent, commutative and zero-divisor free semirings. In particular, we provide improvements of the well-known determinization method of weighted automata over tropical semirings given by Mohri [Computational Linguistics 23 (2) (1997) 269-311].

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