TITLE: Sum and product of ultrafilters
QUESTION [5 upvotes]: can anyone tell me, please, two ultrafilters such that $\mathcal{U}\otimes\mathcal{V}\neq\mathcal{V}\otimes\mathcal{U}$ and others two such that $\mathcal{U}\oplus\mathcal{V}\neq\mathcal{V}\oplus\mathcal{U}$?. Thanks

REPLY [2 votes]: For $\mathcal U\otimes \mathcal V$, let $\mathcal U=(0)$ (principal), $\mathcal V$ -- any other. Then $A=\mathbf N\times \lbrace 0\rbrace$ is not in $\mathcal U\otimes \mathcal V$, (since all vertical sections of $A$ are $\lbrace 0\rbrace\notin \mathcal V$), but is in $\mathcal V\otimes \mathcal U$ (since all vertical sections of $A$ contain $0$).