THE BEST SOLUTION FOR INEQUALITIES OF A O CROSS X LOWER THAN X FROM B O DOT X USING HIGH MATRIX RESIDUATION OF IDEMPOTENT SEMIRING

Abstract A complete idempotent semiring has a structure which is called a complete lattice. Because of the same structure as the complete lattice then inequality of the complete idempotent semiring can be solved a solution by using residuation theory. One of the inequality which is explained is where matrices A,X,B with entries in the complete idempotent semiring S. Furthermore, introduced dual product , i.e. binary operation endowed in a complete idempotent semirings S and not included in the standard definition of complete idempotent semirings. A solution of inequality can be solved by using residuation theory. Because of the guarantee that for each isotone mapping in complete lattice always has a fixed point, then is also exist in a complete idempotent semirings. This of the characteristics is used in order to obtain the greatest solution of inequality . Keywords: complete lattice, complete idempotent semiring, dual Kleene Star, dual product, residuation theor