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Three Methods for Solving Generalized Sylvester Matrix Equations Over Boolean Algebra
Behnam Hashemi, Mahsa Nasrollahi Shirazi and Hanieh Tavakolipour

We consider the generalized Sylvester matrix equation

A X B ∪ C X D = F

over the two-element Boolean algebra.We provide a necessary and sufficient condition for this equation to be solvable. We provide three different proofs for our main result, which in turn lead to three methods for computing the greatest element in the set of all solutions. One of our formulae reduces the computational cost of solving this matrix equation from sixtic to cubic order. An application is given in a decision problem.

Keywords: Boolean algebra, greatest solution, Sylvester matrix equations, Kronecker product.

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