The completeness problem in partial hyperclones

The composition-closed sets of partial multi-valued operations, called partial hyperclones, defined on the finite set E(k)={0,1,…,k-1}(k⩾2) are investigated. It is shown that the lattice of all partial hyperclones is dually atomic, i.e., any non-full partial hyperclone is contained in a maximal partial hyperclone. Based on it some completeness criteria in the full partial hyperclone are established. Next the total list of maximal restriction-closed partial hyperclones is obtained and, thus, the completeness problem with respect to compositions and restrictions of partial hyperoperations is solved.