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In this work, we provide a necessary and sufficient condition on a polyominoideal for having the set of inner 2-minors as degree reverse lexicographicGr\"obner basis, due to combinatorial properties of the polyomino itself.Moreover, we prove that when the latter holds the ideal coincides with thelattice ideal associated to the polyomino, that is the ideal is prime. As anapplication, we describe two new infinite families of prime polyominoes.

Primality of polyomino ideals by quadratic Gr\"obner basis