  Let $G$ be an acylic directed graph. For each vertex $g \in G$, we define an
involution on the independent sets of $G$. We call these involutions flips, and
use them to define a new partial order on independent sets of $G$.
  Trim lattices generalize distributive lattices by removing the graded
hypothesis: a graded trim lattice is a distributive lattice, and every
distributive lattice is trim. Our independence posets are a further
generalization of distributive lattices, eliminating also the lattice
requirement: an independence poset that is a lattice is always a trim lattice,
and every trim lattice is the independence poset for a unique (up to
isomorphism) acyclic directed graph $G$. We characterize when an independence
poset is a lattice with a graph-theoretic condition on $G$.
  We generalize the definition of rowmotion from distributive lattices to
independence posets, and we show it can be computed in three different ways. We
also relate our constructions to torsion classes, semibricks, and
2-simpleminded collections arising in the representation theory of certain
acyclic finite-dimensional algebras.
