  We initiate the study of a large class of species monoids and comonoids which
come equipped with a poset structure that is compatible with the multiplication
and comultiplication maps. We show that if a monoid and a comonoid are related
through a Galois connection, then they are dual to each other. This duality is
best understood by introducing a new basis constructed through M\"obius
inversion. We use this new basis to give uniform proofs for cofreeness and
calculations of primitives for the Hopf monoids of set partitions, graphs,
hypergraphs, and simplicial complexes.
  Further, we show that the monoid and comonoid of a Hopf monoid are related
through a Galois connection if and only if the Hopf monoid is linearized,
commutative, and cocommutative. In these cases, we give a grouping-free formula
for the antipode in terms of an evaluation of the characteristic polynomial of
a related poset. This gives new proofs for the antipodes of the Hopf monoids of
graphs, hypergraphs, set partitions, and simplicial complexes.
