How many identities?

Algebra Level 2

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?

two; two; one infinitely many; zero; zero one; one; one zero; zero; zero zero; three; zero

1 solution

展豪張
Jun 23, 2016

If there exists right identity(s), there is at most 1 left identity.

Proof:

Let $l_1,l_2$ be left identities, and $r$ be a right identity.
$\;\;\;\;l_1$
$=l_1r$ (definition of right identity)
$=r$ (definition of left identity)
$=l_2r$ (definition of left identity)
$=l_2$ (definition of right identity)

The proof of the other side is similar.

So the combination two, two, one is impossible.

How many identities?

1 solution

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