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Technically how? In a way that isn't abstract to either parties in the transaction?

Basically you split your bitcoin and send one part to the recipient and one part back to yourself.

Basically, that is my question: What is the lowest bit of a bitcoin ? For Euro, or other curency is 1/100 aka cent.

1 sat (Satoshi) is 1/100000000 bitcoin, so at the moment it's around 0,001 euro.

And then you get millisatoshis that can be used on the lightning network, but I've not dealt with that directly myself.