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I apologize, I tried to make the title as concise as I could.

If a Blockchain, say the Bitcoin chain, has some NFT that exists uniquely (by definition of an NFT) on that blockchain, what prevents the same NFT from existing on a totally different blockchain, such as the Polygon Blockchain, under new ownership?

In this situation you would have the same digital asset owned by two separate people, on two separate blockchains. With the digital asset being unique only in its own blockchain.

Is there some sort of communication between all blockchains that exist that prevents this from happening?

I may be (probably) not understanding Blockchain technology properly, and I am trying to find the hole in my understanding of the topic. I've tried researching similar topics, but can't seem to find the answers I am looking for. Maybe I am using the wrong terms?

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You're not misunderstanding anything, and there is nothing that prevents that.

Value of these items is similar to art: you trust the author/reproducer not to misrepresent how many copies exist, but there is nothing to prevent them from doing that (regardless of whether that happens on one chain or on multiple), except perhaps their reputation or loss of value of the tokens if discovered. If someone were to issue two tokens, those are two separate digital items, and they can have separate ownership.

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    In a way, the same thing happens when a forkcoin spins off. The whole UTXO set is replicated on the new altcoin, but they are still distinct in the context of the network that they're tracked on.
    – Murch
    Dec 2, 2021 at 15:57
  • Thank you! This question has been bugging me for weeks and it is a relief to know I am not misunderstanding anything. Dec 2, 2021 at 15:57
  • @Murch Thanks for the additional context, that makes a lot of sense. Dec 2, 2021 at 15:59

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