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Yes, of course. Exploiting price differences between exchanges is arbitrage. It's very easy to take advantage of (if nobody else is doing so) so is done early on in any market.

Sounds like you are describing the NBBO (National Best Bid and Offer) which literally requires everyone to execute at the best price (or better) to buy or sell in the market.

If you send an order to an exchange (say NYSE) and they don't have the Best Bid then they will route your order to the other exchanges (NASDAQ/BATS/etc). Is this what you are getting at?

This is what everyone in capital markets execution has been doing for over a decade actually.