# Diagrammatic (Postfix) Composition of Functions

Consider the functions $f : X \to Y$ and $g : Y \to Z$. According to the Wikipedia articles on Function Composition, the application of $f$ to an input $x$ can be written as $xf$ (as opposed to the usual $f(x)$), and function composites can be written as $fg$ (as opposed to the usual $g \circ f$). This is known as postfix notation or diagrammatic notation because the equation $(xf)g = x(fg)$ holds and the function composite can be read from the following diagram: $$X \xrightarrow{f} Y \xrightarrow{g} Z \implies X \xrightarrow{fg} Z$$

I would like a journal reference that uses this notation, preferably briefly explaining its advantages.

What I've tried

The closest reference I have is "Z Notation", where relations $R \subseteq X \times Y$ and $S \subseteq Y \times Z$ can be composed in diagrammatic order using a "fat semicolon":

This use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in Category theory.

Could I also have a journal reference that uses Z notation?

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The notation with ";" is one possible choice. Or you could simply use juxtaposition, as in the diagram. There was a similar question math.stackexchange.com/questions/258344/… and you might find some answers helpful. A good case in favour of postfix composition is made here: iti.cs.tu-bs.de/TI-INFO/koslowj/RESEARCH/RPN – Marc Olschok Feb 27 '13 at 1:27