  We establish negative results about "rectangular" local bases in compacta.
For example, there is no compactum where all points have local bases of cofinal
type \omega x \omega_2. For another, the compactum \beta\omega has no
nontrivially rectangular local bases, and the same is consistently true of
\beta\omega \ \omega: no local base in \beta\omega has cofinal type \kappa x c
if \kappa < m_{\sigma-n-linked} for some n in [1,\omega). Also, CH implies that
every local base in \beta\omega \ \omega has the same cofinal type as one in
\beta\omega.
  We also answer a question of Dobrinen and Todorcevic about cofinal types of
ultrafilters: the Fubini square of a filter on \omega always has the same
cofinal type as its Fubini cube. Moreover, the Fubini product of nonprincipal
P-filters on \omega is commutative modulo cofinal equivalence.
