There is this operation you learn in algebraic topology when working with homotopy groups and loops i.e. paths on a topological space $X$, $p:[0,1]\rightarrow X$ with $p(0) = p(1)$. Basically it is an operation $*$ between two loops $p,q:[0,1]\rightarrow X$ defined by

$$ p*q(t):= \left\{ \begin{array}{ll} p(2t) & t\in[0,\frac{1}{2}]\\ q(2t-1) & t\in(\frac{1}{2}, 1]\\ \end{array} \right. $$

It so happened that I work a lot now with paths on a specific topological space (mostly with real manifolds). And I tend to work a lot with this operation not on loops but on paths in general (i.e. $p(0)$ is not necessarily the same as $p(1)$ but...). I redefined this operation on generic paths in such a way that you get the same equality as above if $p(1) = q(0)$ and otherwise we define $p*q = p$ (so in general this is nothing but a concatenation of paths if this can be done otherwise we leave it to be the first path). With this it is easy to see that the set of paths on $X$ and the operation makes a noncommutative semigroup and I am not sure if there is anything more algebraic about it.

I am not an algebraic topologist so obviously I am just going by intuition here and not any past experience. So my question is: is this something of any relevance to people in algebraic topology (or any other field)?