Galois connection for hyperclones

Abstract

This paper is inspired by the paper of Tarasov [17] in which he investigates maximal partial clones on a two-element set. It happens that the approach of Tarasov can be translated into the language of hyperclone theory. He introduced a notion of quasicomposition which assigns to extended hyperoperations extension of their composition. We introduce a new operation in the set of extended hyperoperationand define a quasiclone as a composition closed set of extended hyperoperations containing all projections which is closed with respect to the new operation. For a Galois connection between sets of extended hyperoperations and power relations, we prove that the set ePolR of all extended hyperoperations e-preserving every relation ρ ∈ R is a quasiclone and that each quasiclone is of the form ePolR for a set R of relations on the power set of A without empty-set. Finally, we re-state results of Tarasov in hyperclone framework. © 2010 IEEE.