Groupoids and labelled quotient graphs: a topological analysis of the modular structure in pyroxenes.

The analysis of the modular structure of pyroxenes, recently discussed in Nespolo & Aroyo [Eur. J. Mineral. (2016), 28, 189-203], has been performed on the respective labelled quotient graphs (LQGs). It is shown that the structure and maximum symmetry of the module, i.e. its layer group, can be determined directly from the LQG. Partial symmetry operations between different modules have been associated with automorphisms of the quotient graph that may not be consistent with net voltages over the respective cycles. These operations have been shown to generate the pyroxene groupoid structure.