Clones are structures whose elements are named operations with the fundamental properties (1) that a composition of operations is defined and (2) that special operations named projections are contained.
Clones on a given carrier set A can be ordered by inclusion to form a complete lattice, where every non-bottom element lies above an atom (minimal clone), provided the carrier set is finite. Each minimal clone must be generated by a single function, and the possible types of minimum arity generators were described in the form of a necessary condition in a famous theorem by Ivo Rosenberg in the 1980s. Despite this fact, the lattice of clones is so complicated that even all minimal clones on A are understood only for |A|<=3 by work of Béla Csákány (1980s) and Emil Post (1920-40s)...
The talk will report on how these new results were obtained (jointly with Edith Vargas-García and Andreas Wachtel, ITAM) and, if time permits, will give an outlook towards future steps.