It seems like email is at least analogous to public-key cryptography, the public portion of the key being your email address, and the private portion of the key being the password to the email account. Is this merely an analogy, or is email in fact an instance of public-key cryptography?
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This is a very abstract and wrong analogy and I assume you are talking about encryption. Often for introducing public key encryption one encounters the "box with lock" example: You publish a box and a lock (your public key) and everyone can put something in the box and close the lock and send it to you. You (the only guy that has the key to the lock - the private key) can unlock the lock and get the stuff that has been put in there, but no one else ( because you are the only guy with the key to the lock) can do so.
But your email example lacks the main desired feature of this analogy (which is called confidentiality), i.e., anyone who runs your email server or listens in the wire can read all the mails intended to you without the need to know your password. Public key encryption relates the public and private key in a mathematical way (by means of some hard to compute function) such that decryption without knowledge of the private key is computationally infeasible, which is obviously not the case in your example.
Interestingly, identity based encryption does something that comes near to what you describe although cryptographically secure. There your email address (or any arbitrary string) can be your public key. But your private key is not an arbitrary independently chosen password, but some secret that is derived from some trusted party based on your email address. Then, anyone can use your email address to encrypt messages for you and only you knowing the corresponding private key can decrypt.