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This is the second introductory paper concerning structures called rootoidsand protorootoids, the definition of which is abstracted from formal propertiesof Coxeter groups with their root systems and weak orders. The ubiquity ofprotorootoids is shown by attaching them to structures such as groupoids withgenerators, to simple graphs, to subsets of Boolean rings, to possibly infiniteoriented matroids, and to groupoids with a specified preorder on each set ofmorphisms with fixed codomain; in each case, the condition that the structuregive rise to a rootoid defines an interesting subclass of these structures. Thepaper also gives non-trivial examples of morphisms of rootoids and describes(without proof, and partly informally) some main ideas, results and questionsfrom subsequent papers of the series, including the basic facts about principalrootoids and functor rootoids which together provide the raison d'\^etre forthese papers.

Groupoids, root systems and weak order II