A question on minimal idempotent ultrafilter on N^2 - MathOverflow

A question on minimal idempotent ultrafilter on N^2

2012-10-07T13:47:23Z

Is there some minimal idempotent ultrafilter $q \in \beta( \mathbb{N}^2)$ (with respect to the law $"+"$) such that any $A \in q$ is a subset of $\mathbb{N} \times { 0 } $ ? (See for example http://www.math.osu.edu/~bergelson.1/vbkatsiveli20march03.pdf for definitions).

Motivation : Van der Waerden theorem can be quickly deduced from a negative answer to this question.

Answer by Andreas Blass for A question on minimal idempotent ultrafilter on N^2

2012-10-07T13:57:04Z

Since $(\mathbb N-\{0\})^2$ is a 2-sided ideal in $\mathbb N^2$ , it follows that $\beta((\mathbb N-\{0\})^2)$ is a 2-sided ideal in $\beta(\mathbb N^2)$ and therefore contains all of the latter semigroup's minimal idempotents.