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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 $ω\timesω$ whose defining characteristic is that the sets of the form $X\times Y$, where $X,Y$ are infinite subsets of $ω$, are dense in the quotient $\mathcal{P}(ω\timesω)\slash\mathcal{NC}$. This quotient has countable partitions and consistently has partitions of size $ω_1$ while $\mathfrak{a}>ω_1$. 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.