Paths on a surface 
Identity element
When we say that the real numbers have an identity element for its product, we are saying that there is a number such that when multiplied (on the right or left) by any other number it doesn't change this number. That is, there is a real number \(e\) such that \(e.a=a.e=a\), for any real number \(a\). For the usual product of real numbers we know that \(e=1\).
On the product of two paths (representing homotopy classes) the reasoning is analogous: observe that there are classes of paths - the constant paths - that when multiplied - properly! - on the left or right by any path we obtain a path homotopic to this path.
Formally, for each point of the surface, the constant path defined by that point is the identity element, respectively on the left or on the right, to the paths that start or end at that point. Then, if \(f\) from \([0,1]\) to \(S\) connects \(A\) to \(A'\), and \(e\) and \(e'\) are the paths of constant with respect to \(A\) and \(A'\), \(e.f\) and \(f.e'\) are homotopic to \(f\).
Taking into consideration the description for the product on paths covered in one hour, the products above will differ on the time and velocity they are covered.
In the case of \(e.f\):
Corresponds to covering \(e\) in the first half an hour and \(f\) in the last one. As \(e\) is a constant path (equal to its starting point \(f\)), this means staying still at the starting point for half an hour and then cover the path \(f\) (at twice the velocity of the original)).
Formally, \(g_{1}=e.f\) from \([0,1]\) to \(S\) can be defined by:\[g_{1}(t)=\begin{cases} \begin{array}{lcc} f(0) & \mbox{if} & 0\leq t\leq0.5\\ f(2t-1) & \mbox{if} & 0.5<t\leq1 \end{array}\end{cases}\]
In the case of \(f.e'\):
Corresponds to covering \(f\) in the first half an hour and \(e'\) in the last. As \(e'\) is a constant path (equal to the end point of \(f\)), this means covering the entire path of \(f\) (at twice the velocity) in the first half an hour and then stay still at the end point in the remaining half an hour.
Formaly, \(g_{2}=f.e'\) from \([0,1]\) to \(S\) can be defined by:\[g_{2}(t)=\begin{cases} \begin{array}{lcc} f(2t) & \mbox{if} & 0\leq t\leq0.5\\ f(1) & \mbox{if} & 0.5<t\leq1 \end{array}\end{cases}\]
Therefore, the product paths are distinct and also distinct from \(f\)! Nevertheless, we can show that these paths are all homotopy equivalent, for this it suffices to continuously change the velocity (or of the coloring of the corresponding rubber band).
Formaly, an homotopy \(H\) from \([0,1]\times[0,1]\) to \(S\) (between \(g_{2}\) and \(f\)) can be given by: \[H(t,x)=\begin{cases} \begin{array}{lcc} f((1-x)2t+xt) & \mbox{if} & 0\leq t\leq0.5\\ f((1-x)+xt) & \mbox{if} & 0.5<t\leq1 \end{array}\end{cases}\]
