Organic and tight

We define organic sets and organically stationary sequences, which generalize tight sets and tightly stationary sequences respectively. We show that there are stationary many inorganic sets (Theorem 3) and stationary many sets that are organic but not tight (Theorem 4). Working in the Constructible Universe, we give a characterization of organic and tight sets in terms of fine structure (Theorem 7). We answer a related question posed in [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: Part two, Ann. Pure Appl. Logic 142 (2006) 55–75] about the combinatorial principle Coherent Squares (Corollary 9).