Small infinite partitions and other features of the Nowhere Centered ideal

The \textit{nowhere dense ideal} $\mathcal{NC}$ is introduced. It is a coanalytic ideal of $\omega\times\omega$ whose defining characteristic is that the sets of the form $X\times Y$, where $X,Y$ are infinite subsets of $\omega$, are dense in the quotient $\mathcal{P}(\omega\times\omega)\slash\mathcal{NC}$. This quotient has countable partitions and consistently has partitions of size $\omega_1$ while $\mathfrak{a}>\omega_1$. This represents a huge contrast with other definable quotients. Other combinatorial features of this ideal are presented, as well as some results on a family of similar, higher dimensional ideals.