How many identities?

Algebra Level 3

Suppose \(S\) is a set with a binary operation. Consider the following sentence about the identity elements in \(S\):

\(S\) has \(\underline{\phantom{1234567}}\) left identities, \(\underline{\phantom{1234567}}\) right identities, and \(\underline{\phantom{1234567}}\) identity elements.

Which choice of words for the blanks gives a sentence that cannot be true?

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