On the partial hyperclone lattice

Abstract

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 hype rope rations. It is proved in [5] that the mapping λ from the lattice of partial hyperclones on A into the lattice of clones of operations on P(A) defined by λ(C) = δ(C#), where δ(C#) is the clone of operations on P(A) generated by C # , 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 [λ( J A ), λ(Hp A )] but these are not in the set imλ of all images of the mapping λ, where J A is the set of all (partial) hyperprojections and HpA is the set of all partial hyperoperations on A. © 2005 IEEE.