I'm doing some philosophy involving time evolution operators. Thus I might have two operators $A$ and $B$ which operate in order on the state of the world, $x$. This might be written mathematically as $B(A(x))$ or in the multiplicative notation for groups and semigroups, $BAx$.
For time evolution, it is intuitive to write $xAB$, as time evolves from left to right. This is intuitive purely because of culture. Obviously both are equally valid. I merely need to define the operators with reversed multiplication tables.
Why do we choose to define our transformation operators to always appear on the left side of the multiplication? Is it merely because that's the order they appear in function composition, or is there something deeper?
The convention that operators go on the left is not universal. For example, Herstein's "Topics in Algebra" is written with operators on the right. See e.g. this