# About left / right Inverses

Just out of curiosity

if I have the function F and G. So that G is the left inverse of F. Because G o F = Idx.

My question is why is it named "left" inverse if it is like that? Is it because of the position of G in the composition notation in relation to F (G is at the left of F in the G o F = Idx equation)?

Another possible guess would be :

it's called a "left " inverse of F because to do G o F = Idx means that it goes from the domain to the codomain, then back to the domain again (the domain is usually illustrated at the left)

It's called a "right" inverse F because to do F o H = Idy means that it goes from the codomain to the domain, then back to the codomain again (the codomain is usually illustrated at the right side).

Or is it named "left/right" inverse because of some other reason? Does anybody knows the answer?

Regards.

• I think your first guess is right. For example, in a semigroup with unit $e$, we say $a$ is a left inverse of $b$ if $ab=e$. It refers to the fact that $a$ is on the left when the product is written. – Gerald Edgar May 20 '16 at 13:22

## 1 Answer

Functions $F$ and $G$ have different domains, say $D_F=A$ and $D_G=B$. So there are two identity maps $I_A$ and $I_B$. Thus the relations $FG=I_B$ and $GF=I_A$ are very different things. In the first case we say that $G$ is the right inverse of $F$ and in the second case it is the left inverse. Of course the terminology has the evident origin: on which side of $F$ $G$ stands in the identity. The properties of $F$ to be right-invertible and left-invertible are different properties.

• UI'm not so clear about "on which side of FG stands in the identity". Doesn't that also depends on the inverse of which function you are talking about? F or G. Can you give me an example of this. – Gin99 May 21 '16 at 0:45
• In the identity $FG=I$, $F$ stands on the left and $G$ on the right. In the identity $GF=I$ vise versa. – Alexandre Eremenko May 21 '16 at 3:55
• Let me verify, so you mean my first guess : That is, G is called "left / right" inverse of F because of it's (G) position in the composition notation (G o F = Id or F o G = Id) relatively to F. Correct? – Gin99 May 21 '16 at 5:26
• Yes, your initial guess was correct. – Alexandre Eremenko May 21 '16 at 13:25
• You said "Of course the terminology has evident origin...", can you tell us how did you come about this? – Gin99 May 21 '16 at 22:30