Minimal primes of ideals arising from conditional independence statements

We study ideals whose primary decomposition specifies the relevant structural zeros of certain conditional independence models. The ideals we study generalize the class of ideals considered by Fink (2011) [5] in a way distinct from the generalizations of Herzog, Hibi, Hreinsdottir, Kahle, and Rauh (2010) [10] and Ay and Rauh (2011) [1]. We introduce switchable sets to give a combinatorial description of the minimal prime ideals, and for some classes we describe the minimal components. We discuss possible interpretations of the ideals we study, including as 2×22×2 minors of generic hypermatrices. We also introduce a definition of diagonal monomial orders on generic hypermatrices to compute some Gröbner bases.