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# MINIMAL CLONES WITH ABELIAN REPRESENTATIONS

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MINIMAL CLONES WITH ABELIAN REPRESENTATIONS KEITH A KEARNES Abstract We show that a minimal clone has a nontrivial abelian representation if and only if it is isomorphic to a minimal subclone of a finite cyclic group As an application we show that a minimal clone contains a Mal cev operation if and only if it is isomorphic to the clone of idempotent operations of a group of prime order 1 Introduction A clone is trivial if every operation is projection onto a variable A clone is minimal if it is not trivial but its only subclone is trivial Any minimal clone is generated by a single operation If C is a clone generated by the operation f then an f representation of C is a pair A hA f A i where A is a set f A is an operation on A and the assignment f 7 f A extends to a clone homomorphism from C to the concrete clone of operations on A generated by f A Such a representation will be called trivial if the clone of hA f A i is a trivial clone i e if f A is projection onto a variable A representation is faithful if the assignment f 7 f A extends to a clone isomorphism from C to the clone of operations on A generated by f A The size of the representation hA f A i is A Let C be a minimal clone and let f be an operation generating C The class of f representations of C is a variety of algebras which we denote by V The clone of V is C which is a minimal clone by assumption and the clone of each A V is either trivial or it is a minimal clone If we change from f to a different operation generating C then we get a term equivalent variety In this paper we shall often drop the reference to f when we speak of representations and consequently we shall only consider algebras and varieties up to term equivalence The term condition for an algebra A cf Chapter 3 of 3 is the assertion that for all operations t x y in Clo A and all a b A u v An the following implication holds t a u t a v t b u t b v 1 2 KEITH A KEARNES An abelian algebra is one which satisfies the term condition When using

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