  Let $C \subset {\bf N}^d$ be an affine semigroup, and $R=K[C]$ its semigroup
ring. This paper is a collection of various results on "$C$-graded"
$R$-modules, especially, monomial ideals. For example, we show the following:
If $R$ is normal and $I$ is a radical monomial ideal (i.e., $R/I$ is a
generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay
property of $R/I$ is a topological property of the "geometric realization" of
the cell complex associated with $I$. Moreover, we can give a squarefree
modules/constructible sheaves version of this result.
