Idempotent matrices over antirings

We study the idempotent matrices over a commutative antiring. We give a characterization of idempotent matrices by digraphs. We study the orbits of conjugate action and find the cardinality of orbits of basic idempotents. Finally, we prove that invertible, linear and idempotent preserving operators on n×n matrices over entire antirings are exactly conjugate actions for n⩾3. We also give a complete characterization of the 2×2 case.