What is a 2-sided inverse?

Question

The text says:

For every permutation $\sigma$ there is a 2-sided inverse function $\sigma^{-1}: \Omega \to \Omega$ satisfying $\sigma \circ \sigma^{-1} = 1$.

So I am wondering - what is the 2-sided inverse function...?

Answer 1

If I give you an arbitrary function $f:A\to B$ then it might turn out there exists $h:B\to A$ such that $f\circ h={\rm id}_B$ or $g:B\to A$ such that $g\circ f={\rm id}_A$. In such a case we say $h$ is a right inverse, and that $g$ is a left inverse. It might happen that there exists a $j$ which is both a right and a left inverse for $f$, and I such case we merely say $j$ is an (actually the) inverse of $f$, and denote it by $f^{-1}$. In particular, injective functions are those who have left inverses, or which are said to be left cancellable ($fh=fg\implies h=g$), and surjective functions are those who have (AOC) right inverses, or which are said to be right cancellable ($gf=hf\implies g=h$).

Answer 2

"Two-sided" means that

$$\sigma \circ \sigma^{-1} = id \text{ and } \sigma^{-1} \circ \sigma = id$$

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In the case of a permutation, the inverse is defined via

$$\sigma^{-1} (y) = x \iff \sigma(x) = y$$

This is well-defined precisely because $\sigma$ is a bijection.

Answer 3

In addition to the first answer (+1) I want to add that is not always assured the existence of a two sided inverse for an element of a set together with an operation.

just think about an only injective function from a set to another or an only surjective function between the same two set. They have only a 1-sided inverse (find out why).

And you can continue proving that a function is a bijection if and only if it has a two-sided inverse (just combine the two facts above). In fact permutations are the bijective function from a set to the same set.

In order to have some nice properties sometimes we need to have a two sided inverse, for example the fact that a two,sided inverse is unique (a 1-sided inverse need not to be unique, just check the first case I show you about only injective or surjective function,they have many different inverses :) )