  The defining characteristic of an exceptional point (EP) in the parameter
space of a family of operators is that upon encircling the EP eigenstates are
permuted. In case one encircles multiple EPs, the question arises how to
properly compose the effects of the individual EPs. This was thought to be
ambiguous. We show that one can solve this problem by considering based loops
and their deformations. The theory of fundamental groups allows to generalize
this technique to arbitrary degeneracy structures like exceptional lines in a
three-dimensional parameter space. As permutations of three or more objects
form a non-abelian group, the next question that arises is whether one can
experimentally demonstrate this non-commutative behavior. This requires at
least two EPs of a family of operators that have at least 3 eigenstates. A
concrete implementation in a recently proposed $\mathcal{PT}$ symmetric
waveguide system is suggested as an example of how to experimentally check the
composition law and show the non-abelian nature of non-hermitian systems with
multiple EPs.
