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Harming someone’s reputation in speech with falsehoods is known as slander, and doing the same thing in writing is known as libel (which sometimes includes speech as well). Any ordinary citizen who can claim to have suffered harm as a result of such defamation may sue.

A general offer is one that is made to the public at large. It is not made any specified parties. So any member of the public can accept the offer and be entitled to the rewards/consideration. Say for example you put out a reward for solving a puzzle.