Ultrafilter subtraction and "zero"

Question

This is related to a couple recent MO/MSE questions of mine, namely 1,2. Belatedly, I've tweaked this post to remove an overly-ambitious secondary question; see the edit history if interested.

Let $\beta\mathbb{Z}$ be the set of all ultrafilters on $\mathbb{Z}$, and as usual conflate $n$ and $\{A\subseteq\mathbb{Z}:n\in A\}$ for $n\in\mathbb{Z}$. We can extend any binary operation on $\mathbb{Z}$ to a semicontinuous analogue on $\beta\mathbb{Z}$, at the cost of many (most?) algebraic properties. Ultrafilter addition is quite well studied (see e.g. Hindman/Strauss), but I've been able to find much less about ultrafilter subtraction: $$\mathcal{U}\widehat{-}\mathcal{W}=\{A\subseteq\mathbb{Z}: \{k:\{a: a-k\in A\}\in\mathcal{U}\}\in\mathcal{W}\}.$$

Say that an ultrafilter $\mathcal{W}$ is zeroid iff $\mathcal{W}=\mathcal{U}\widehat{-}\mathcal{U}$ for some $\mathcal{U}$. My question is:

Which ultrafilters are zeroid?

To make this actually answerable, I tentatively guess that $(i)$ $0$ is the only zeroid principle ultrafilter but not the only zeroid ultrafilter and $(ii)$ $p$-points are not zeroid; are these guesses true?

Answer 1

Both your guesses are correct. To see this, it's helpful to reformulate the way you're thinking about the subtraction operator on $\beta \mathbb Z$. Beginning with subtraction on $\mathbb Z$, you can first extend this to an operator $\beta \mathbb Z \times \mathbb Z \rightarrow \beta \mathbb Z$ by setting $\mathcal U - n = \{B-n :\, B \in \mathcal U\} = \{A \subseteq \mathbb Z :\, A+n \in \mathcal U\}$. Notice that this agrees with your definition of subtraction on $\beta \mathbb Z$ when we identify $n$ with the principle ultrafilter at $n$. So this is the "right" way to think of subtracting an integer from an ultrafilter. But then there is only one way to extend this to a semi-continuous operation $\beta \mathbb Z \times \beta \mathbb Z$: we must define $$\mathcal U - \mathcal W = \textstyle {\mathcal W}\text{-}\!\lim_n (\mathcal U-n).$$ This again agrees with your definition (it must!), but personally I find it much more intuitive to think of $\mathcal U - \mathcal W$ as a topological limit of the sequence $\mathcal U, \mathcal U - 1, \mathcal U - 2, \dots$.

This description of what's going on makes your two guesses plainly true. For $(i)$, note that if $\mathcal U$ is nonprincipal then so is $\mathcal U - n$ for all $n$, which means (because $\mathbb Z$ is open in $\beta \mathbb Z$) that the limit ${\mathcal W}\text{-}\!\lim_n (\mathcal U-n)$ is in $\beta Z \setminus \mathbb Z$ for any $\mathcal W$ (including $\mathcal W = \mathcal U$). For $(ii)$, note that ${\mathcal W}\text{-}\!\lim_n (\mathcal U-n)$ is a limit of a countable sequence of points in $\beta \mathbb Z \setminus \mathbb Z$, hence not a P-point. In fact, let me observe that $\mathcal U - \mathcal W$ is never a weak P-point (for the same reason), which shows that not every ultrafilter is representable as the difference of two others.