I am working with the notion of Weyl Groupoids as introduced in "A generalization of Coxeter groups, root systems, and Matsumoto’s theorem" by Heckenberger and Yamane.
The authors generalize the connection between Weyl groups and its root systems to Weyl groupoids and its generalized root systems, but I cannot convince myself that for any given Weyl groupoid of some type C (with regard to the Cartan scheme C) there always is a generalized root system of this type to begin with.
While tied to an already existing root system of type C, their definition of generalized real roots should for any choice of bases emit a root system of type C I think … but I cannot convince myself enough to be sure. Any insights would be well appreciated.
Oh, and please excuse my code free and therefore lack of explicit definitions question. I hope the question itself is clear enough!