Goal: A mathematical hierarchy is an ordered set of elements in which the ordering reflects a natural hierarchical relation between (or arrangement of) the elements of the set. A partially ordered set (a ‘poset’) is a set in which, for a pair of elements, one element precedes the other according to some criterion that compares the two elements. Each element is associated with some property; elements with the same property are grouped together (and are often called a class). It is common that such classes (of elements with equivalent properties) are organized in terms of increasing complexity, hence the term ‘hierarchy’ (Hawthorne, F.C. (2014) The structure hierarchy hypothesis. Mineralogical Magazine, 78, 957-1027).

A structure hierarchy is a set of structures ranked according to the polymerization of coordination polyhedra of higher bond-valence from lower to higher connectivity. Such hierarchies (1) provide a framework to understand the factors controlling composition and structural variability of minerals (e.g., borates: Schindler & Hawthorne 2001a,b,c); and (2) help link particular chemical compositions and structural arrangements to specific mineral parageneses.