Semigroups and units

Question

I am working my way though Basic Algebra 1. I am currently on chapter 1 , and more specifically I busy with the following exercise:

"Let $S$ be a set a define a product in $S$ by $ab=b $. Show that $S$ is a semigroup. Under what condition does $S$ contains a unit"

I just have one question about this exercise. It concerns the notion of an unit. I am correct in assuming that the term "unit" is a synonym for identity?

Answer 1

In abstract algebra, a "unit" (or "unity") can refer to the identity element, but it can also mean an element $a$ such that there exists an element $b$ satisfying $ab=ba=e$ where $e$ is an identity element.

Fortunately, no matter which interpretation you use, the exercise will wind up at the same conclusion.

Suppose the semigroup has an identity $e$: you should be able to draw a very strong conclusion about the entire semigroup. Keep in mind that the definition of an identity requires that $ae=ea=a$ for all $a$.

Since you have to have an identity $e$ before you can talk about "units" $ab=ba=e$, you are already taking care of the other (less likely) interpretation. I mean that if you assume there exists $e,a,b$ such that $e$ is the identity and $ab=ba=e$, then you are already in the situation in the grey box above.

Answer 2

Units are invertible elements in the semi-group which is a monoid.