How many idempotent ultrafilters (under addition) are there, in terms of cardinality? Also, does the set of idempotents have any nice topological properties?
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See this question. – Dietrich Burde Mar 23 '17 at 19:14
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@Dietrich: Seeing how the linked question was cross-posted to MathOverflow, where the OP has commented asking this very question well before posting this question, I imagine he is aware of that question. – Asaf Karagila Mar 23 '17 at 19:56
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1There are $2^{\mathfrak{c}}$ idempotents in $\beta \omega$ - See Theorem 6.9 in Hindman and Strauss, Algebra in the Stone-Cech compactification: Theory and applications, Walter de Gruyter, Berlin/New York, 1998. – hot_queen Mar 24 '17 at 12:54
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Ashutosh, is this hard to show? – mbsq Mar 25 '17 at 06:24
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I don't think so. I wrote the main steps of their proof. They shouldn't be too hard to check. – hot_queen Mar 25 '17 at 11:53
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Here are the main steps of the proof from the Hindman-Strauss book.
Step 1: Let $A = \{2^n : n \geq 1\}$. Suppose $p, q \in \beta \mathbb{N} \setminus \mathbb{N}$, $A \in p \cap q$ and $p \neq q$. Then $(\beta \mathbb{N} + p) \cap (\beta \mathbb{N} + q) = \phi$. In particular, $\beta \mathbb{N}$ has $2^{\mathfrak{c}}$ pairwise disjoint left ideals.
Step 2: Every left ideal in $\beta \mathbb{N}$ contains a minimal left ideal.
Step 3: Every minimal left ideal in $\beta \mathbb{N}$ contains an idempotent.
It follows that $\beta \mathbb{N}$ has $2^{\mathfrak{c}}$ idempotents.
hot_queen
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The definition of a left ideal here is: $\phi \subsetneq J \subseteq \beta \mathbb{N}$ is a left ideal if $\beta \mathbb{N} + J \subseteq J$. – hot_queen Mar 25 '17 at 14:45
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1Steps 2 and 3 follow from some compactness arguments, if we replace left with right. I have no idea how to do that for left ideals. I think maybe the definition of the operation is switched. – mbsq Mar 25 '17 at 14:47