# Is there a commonly used notation for flipped composition?

We have $(f \circ g) = x \mapsto f(g(x))$ however since I read left to write it always seems backwards to me. Is there a symbol like $( g \ggg f) = x \mapsto f(g(x))$.

It is especially grating in situations like.

$$\require{AMScd} \begin{CD} X @>{f \circ g}>> Y \\ @VVgV @AAfA \\ g(X) @>{id}>> g(X)\end{CD}$$

• I've seen people denote $g(f(x))$ by $(f\circ g)(x)$ in the context of group theory. – Git Gud Oct 23 '16 at 23:16
• That's unpleasantly ambitious :( – Q the Platypus Oct 23 '16 at 23:19
• I think the idea is that $f$ and $g$ appear in the same order in both notations. That's what we get for having left-hand function notation. Were I to go back in time to the ancient Greeks and reboot mathematics, I would establish the notation $(x)f$ (or maybe use something different from parentheses). Along with establishing $6.28$ as the circle constant (I would probably still call it $\pi$, for "perimeter", though), and the electron as positively charged. – Arthur Oct 23 '16 at 23:19
• @QthePlatypus Actually that's not what I've seen. Arthur's comment reminded me of it. What I've seen is $(x)(f\circ g)$ denoting $g(f(x))$. – Git Gud Oct 23 '16 at 23:49
• If you write your functions on the right, $x \mapsto xf$, then $fg$ means first do $f$ then $g$. Too bad the convention of writing functions to the left of their arguments got started a long time ago. – user4894 Oct 24 '16 at 0:41

Computer scientists may write g ; f for f ∘ g

I personally use the notation $f$»$g$, which is inspired by the F# operator >>. Perhaps you like that too, although it isn't established by any means.

In LaTeX it should be \text{\guillemotright} with:

\usepackage[T1]{fontenc}


and

 \usepackage{amsmath}


Some math books (especially abstract algebra) may write composition in the reverse order: $\sigma\tau$ means: first $\sigma$ then $\tau$. For notation they write $$x^{\sigma \tau} = \big(x^\sigma\big)^\tau$$ For example, this may be seen with field automorphisms.