Clonoids between modules

Abstract

Clonoids are sets of finitary functions from an algebra [Formula: see text] to an algebra [Formula: see text] that are closed under composition with term functions of [Formula: see text] on the domain side and with term functions of [Formula: see text] on the codomain side. For [Formula: see text] (polynomially equivalent to) finite modules we show: If [Formula: see text] have coprime order and the congruence lattice of [Formula: see text] is distributive, then there are only finitely many clonoids from [Formula: see text] to [Formula: see text]. This is proved by establishing for every natural number [Formula: see text] a particular linear equation that all [Formula: see text]-ary functions from [Formula: see text] to [Formula: see text] satisfy. Else if [Formula: see text] do not have coprime order, then there exist infinite ascending chains of clonoids from [Formula: see text] to [Formula: see text] ordered by inclusion. Consequently any extension of [Formula: see text] by [Formula: see text] has countably infinitely many [Formula: see text]-nilpotent expansions up to term equivalence.