I think it depends on the mint policy. The cashu protocol seems to be suited to work well on top of Bitcoin/ LN, but you have to trust the ecash mint to really own the underlying Bitcoins

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Okay. Mmh. Okay. But would I then could do the same transaction with someone else, offline .. that uses another mint?

Or do I totally misunderstand this?

Afaik no you couldn't

The mint to mint transfer are done through lightning so you need online devices

Okay, so could you in theory have a city mint. And everyone using this city mint, could go around the city without internet for a whole day and pay each other?