On the partial hyperclone lattice

For any finite set A, the partial clone lattice on A is embedded into the partial hyperclone lattice on A. It is shown that there are maximal intervals in the partial hyperclone lattice and there are four minimal partial hyperclones such that their join contains all partial hyperoperations. It is proved in (T. Drescher et al., 2001) that the mapping /spl lambda/ from the lattice of partial hyperclones on A into the lattice of clones of operations on P(A) defined by /spl lambda/(C)=/spl delta/(C/sup #/), where /spl delta/(C/sup #/) is the clone of operations on P(A) generated by C/sup #/, is an order embedding, but not a full one. In this paper, it is proved that there are continuum many clones of operations on P(A) that are in the interval [/spl lambda/(J/sub A/), /spl lambda/(Hp/sub A/)] but these are not in the set im/spl lambda/ of all images of the mapping /spl lambda/, where J/sub A/ is the set of all (partial) hyperprojections and Hp/sub A/ is the set of all partial hyperoperations on A.