From clones to hyperclones

Jelena Čolić Oravec, Hajime Machida, Jovanka Pantović, Gradimir Vojvodić

This paper is a survey of different approaches to the study of the lattice of clones of hyperoperations, summarising contributions of the authors in the field. We present three embeddings that are suitable for analysis of the lattice of hyperclones and give more details on embedding of the lattice of hyperclones on a finite set into the lattice of clones on its power set. On a two-element set, we give description of its atoms, coatoms, and the interval generated by unary hyperoperations. On a set, we describe four classes of coatoms, determined by four classes of Rosenberg's relations. Finally, we analyse several Galois connections between particular sets of hyperoperations and relations.