1

So I recently asked the question "Is there a commutative operation for which the inverse of the operation is also commutative?", but this made me realize that I don't really know how to properly define an inverse operation. I'm curious if this even can be defined. For one you could define an inverse operation $\overline{\circ}$ to the operation $\circ$ as $a\overline{\circ}b = a\circ b^{-1}$, as in multiplication and addition (or $a\overline{\circ}b = a^{-1}\circ b$ if the order is right-to-left), but is this the only definition? Or are there cases where you have $a\overline{\circ}b = a\circ c,\quad c\neq b^{-1}$, or some other properties?

Community
  • 1
Frank Vel
  • 5,339
  • Given a function $f:S \rightarrow S$ for sets $S$ and $T$, the inverse function $f^{-1}:T \rightarrow S$ is the function such that $f^{-1}(f(x))=x$ for all $x \in S$. However, such an $f^{-1}$ may not exist. – Ducky Feb 13 '15 at 18:09
  • That should be $S \rightarrow T$ in my first comment. As an example, the function from $\mathbb{R}$ to $\mathbb{R}$ which sends $x \in \mathbb{R}$ to $2x$ has an inverse which sends $x$ to $\frac{x}{2}$. – Ducky Feb 13 '15 at 18:17
  • @Ducky Don't you mean that $f^{-1}$ may not be expressible in terms of elementary operations? – Frank Vel Feb 13 '15 at 18:19
  • Not exactly. That may be the case too, but my definition required $f^{-1}$ to be a function, which in particular means that $f^{-1}(x)$ must be unique. Does the phrase "bijection" ring a bell? There is a theorem that $f$ has an inverse if and only if it is a bijection. – Ducky Feb 13 '15 at 18:22
  • @Ducky Ah right I read a bit too quick. You could still define a set of inverse functions then, although that might quickly become impractical... – Frank Vel Feb 13 '15 at 18:25

2 Answers2

1

The problem here is that inverse is suitable to be defined on numbers not on operations.

Consider the case where: $a=0$

Arashium
  • 2,541
1

Let $(R,\cdot)$ be a set with an operation. My idea would be that if $a\cdot b=c$ then $c\cdot^{-1}b=a$, which can be defined for the elements $b$ of $R$ for which the function $f:R\rightarrow R$, $f_b(a)=a\cdot b$ is injective and $c\in Im(f_b)$. (which for example does not hold for $b=0$ and the classical multiplication.)

Iulia
  • 1,306