  A certain squarefree monomial ideal $H_P$ arising from a finite partially
ordered set $P$ will be studied from viewpoints of both commutative algebra and
combinatorics. First, it is proved that the defining ideal of the Rees algebra
of $H_P$ possesses a quadratic Gr\"obner basis. Thus in particular all powers
of $H_P$ have linear resolutions. Second, the minimal free graded resolution of
$H_P$ will be constructed explicitly and a combinatorial formula to compute the
Betti numbers of $H_P$ will be presented. Third, by using the fact that the
Alexander dual of the simplicial complex $\Delta$ whose Stanley--Reisner ideal
coincides with $H_P$ is Cohen--Macaulay, all the Cohen--Macaulay bipartite
graphs will be classified.
