Maximal hyperclones on E² as hypercores

Abstract

The set of clones of operations on {0, 1} forms a countable lattice which was classified by Post. The cardinality of the lattice of hyperclones on {0, 1} was proved by Machida to be of the continuum. A hyperclone C1 is a hypercore of a clone C if its extension C#1 is contained in C and C1 is not contained in any other hyperclone having the same property. We describe six maximal hyperclones on the two-elements set as hypercores of particular clones on three element set. The interval of hyperclones on { 0, 1 } between the clone of all projections and the one generated by all unary hyperoperations is also completely determined. © 2009 Old City Publishing, Inc.