Clones on infinite sets

Abstract

A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of Cl(X) for infinite X. In the first chapter we prove the combinatorial result that if X is linearly ordered, then the median functions of different arity defined by that order all generate the same clone. The second chapter deals with clones containing the almost unary functions, that is, all functions whose value is determined by one of its variables up to a small set. We show that on X of regular cardinality, the set of such clones is always a countably infinite descending chain. The third chapter generalizes a result due to L. Heindorf from the countable to all uncountable X of regular cardinality, resulting in an explicit list of all clones containing the permutations but not all unary functions of X. Moreover, all maximal submonoids of the full transformation monoid which contain the permutations of X are determined, on all infinite X; this is an extension of a theorem by G. Gavrilov for countable base sets.