On polynomial systems of equations in square matrices filled with natural numbers

The positive existential theories of the sets $M_n(\mathbb{N})$ without parameters build an inclusion lattice isomorhic with the lattice of divisibility. All these sets are algorithmically undecidable. In further sections some easier observations are made, like the undecidability of Diophantine equations with coefficients in $M_n(\mathbb{Z})$.