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Writing function application as a left action has been baked into mathematics since the Bernoullis in the 1700s (cf. this MSE question). Because of this tradition, a lot of mathematics notation has been chosen to complement this.

One (perhaps shocking) example is the commutator $[a,b]$ in a group. Traditionally we use left group actions (to be in line with left-function application), and so the conjugation action is $g \cdot h = ghg^{-1}$. This leads us to define $[a,b] = aba^{-1}b^{-1}$ (so that, among other things, $s \cdot [a,b] = [s \cdot a, s \cdot b]$).

It has become accepted to instead use right group actions, denoted $h^g = g^{-1}hg$, and with this convention we would instead write $[a,b] = a^{-1}b^{-1}ab$. This gives the identity $[a,b]^s = [a^s, b^s]$.

There are lots of places in mathematics where we have an arbitrary choice to make. Left/Right Cosets in algebra, Left/Right Linearity of the complex inner product, etc. I am curious if any of these conventions have to do with functions acting on the left or the right, as in the above example.

Thanks in advance!

HallaSurvivor
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  • It is a convention backed in history (as you say): there is no "deep" reason for it and no reason to change it. Probably the motivation is only the Western way of writing from left to right: thus, it seems natural to symbolize "function $f$ applied to argument $x$" with $f:x$ or $f(x)$. – Mauro ALLEGRANZA Oct 10 '19 at 07:17
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    It's also highly related to the convection in linear algebra that we have to consider column vectors for connecting linear maps to (left) matrix multiplication. In contrast, there are algebra books that rigorously put function applications to the right. – Berci Oct 10 '19 at 07:34
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    I think that there is a good argument for adopting a notation in which functions act on the right. For example, if we have $$X \overset{f}{\to} Y \overset{g}{\to} Z, $$ then we have a function $g \circ f$, which reads backwards from the diagram. Related difficulties arise in fractal geometry, where iterated function systems are often of interest: if $\alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n)$ is a multiindex and $f^{\alpha}$ denotes a composition of maps, is $f_{\alpha_1}$ the first map we apply, or the last one? That being said, the acts-on-the left convention is here to stay. ;) – Xander Henderson Oct 10 '19 at 13:17
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    @Mauro I agree with you all! I'm not trying to switch to functions on the right! I just realized that the definitions of conjugation and commutator are based in our (arbitrary) choice of left vs right. I'm curious if there's other examples of this. – HallaSurvivor Oct 10 '19 at 15:15
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    Ordinal multiplication is not commutative. The product $AB$ of ordinals $A, B$ is the ordinal which is order-isomorphic to the lexicographic order on $B\times A$, and the lexicography is English (or German or ...), not Hebrew (nor Arabic ...). Because someone in the 20th century (J. von Neumann?) liked it. – DanielWainfleet Oct 18 '19 at 16:58

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