Chordality, syzygies, and shellability for hypergraphic analogues of interval graphs

Interval graphs are a special class of chordal graphs, and hence have connections to commutative algebra via Fr\"oberg's theorem that characterizes linear resolutions of squarefree quadratic ideals. In recent years, several hypergraphic analogues of interval and chordal graphs have been proposed, in part as an effort to extend Fr\"oberg's theorem to ideals generated in higher degree. In this paper, we study two such classes from the literature, cointerval hypergraphs and underclosed complexes, and show that they are in fact equivalent up to complementation. We then consider their place in the broader theory of higher-dimensional chordality, proving that an underclosed clutter is chordal in the sense of Woodroofe. As a consequence, we answer a question of Dochtermann and Engstr\"om by showing that the associated Alexander dual complexes are vertex decomposable, implying that the corresponding circuit ideals have linear quotients. We furthermore show that these dual complexes have shellings induced by their underclosed vertex orders.