Is the unit in a Monoid unique?

Question

So my proffessor keeps asking me to find the "Units" in monoids and so far I have always proven that there is one and one only.

In general if something is a unit it must satisfy that:

$xi = ix =x$

So if I assume 2 different units $i_1 \neq i_2$

I get that $i_1i_2=i_1=i_2$ which is a contradiction.

So is my proffesor just messing up with me by adding the plural or am I not seeing something important here?

Answer 1

Of course the identity element in a monoid is unique, for the exact reason you stated. A unit $a$ is usually meant to be something that has an inverse; i.e., there is a $b$ such that $ab=ba=i$, where $i$ is the identity in the monoid. So, your professor is probably asking you to find all invertible elements in the monoid.