@Walter Hinteler,
For all I know, legally Northern Ireland might not be able to stay in the EU or even have any EU policy separate from the UK's as a whole. I am unfamiliar with the agreements behind the making of the UK. Maybe Northern Ireland can chart its own course as regards the EU, maybe not. It would not surprise me if the issue is somewhat up in the air, legally.
But whatever position Northern Ireland is in as regards to the EU, the letter from Arlene Foster to the heads of other countries changes nothing. She is no longer in a position of legal power to enforce anything anymore than any other member of the Northern Ireland parliament, she is no longer head of state. This letter implies that she is, or the fact that the position of power she once held is now vacant should be considered a mere formality.
It's really insulting to the heads of other countries, for Arlene Foster seems to think she can pull off the old trick of "Tell everyone you are in charge and maybe they'll be nuts enough to go along with it".