MINIMAL CLONES WITH ABELIANREPRESENTATIONSKEITH A. KEARNESAbstract. We show that a minimal clone has a nontrivial abelianrepresentation if and only if it is isomorphic to a minimal subcloneof a finite cyclic group. As an application, we show that a minimalclone contains a Mal’cev operation if and only if it is isomorphicto the clone of idempotent operations of a group of prime order.1. IntroductionA clone is trivial if every operation is projection onto a variable. Aclone 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 clonegenerated by the operation f , then an f–representation of C is a pairA = hA; fAi where A is a set, fAis an operation on A, and the as-signment f 7→ fAextends to a clone homomorphism from C to theconcrete clone of operations on A generated by fA. Such a representa-tion will be called trivial if the clone of hA; fAi is a trivial clone; i.e., iffAis projection onto a variable. A representation is faithful if the as-signment f 7→ fAextends to a clone isomorphism from C to the cloneof operations on A generated by fA. The size of the representationhA; fAi is |A|.Let C be a minimal clone and let f be an operation generating C. Theclass of f–representations of C is a variety of algebras which we denoteby 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. Ifwe change from f to a different operation generating C, then we get aterm equivalent variety. In this paper we shall often drop the referenceto f when we speak of representations and consequently we shall onlyconsider algebras and varieties up to term equivalence.The term condition for an algebra A (cf. Chapter 3 of [3]) is theassertion that for all operations t(x, ¯y) in Clo(A) and all a, b ∈ A,¯u, ¯v ∈ Anthe following implication holds:t(a, ¯u) = t(a, ¯v) ⇒ t(b, ¯u) = t(b, ¯v).12 KEITH A. KEARNESAn abelian algebra is one which satisfies the term condition. Whenusing the term condition in this paper we shall underline the positionswhere the arguments change in order to make the application moretransparent.Our project is to describe the minimal clones which have nontrivialabelian representations. We approach minimal clones by looking attheir nontrivial abelian representations because of an interesting factabout (nonunary) minimal clones. It may be stated as follows: nontriv-ial abelian representations must be faithful. The classification problemfor minimal clones with a nontrivial abelian representation reduces tothe classification problem for minimal clones of abelian algebras andthis is a problem easily solved.The description of the minimal subclones of a module follows imme-diately from our main theorem (which is Theorem 3.11). However, ourresult is more than this. In the first place, in our arguments we mustallow the possibility that some abelian representations are unrelated tomodules. But even in the case where a clone C has a nontrivial repre-sentation as a reduct of a module, we must worry about what happensif this representation is not faithful. This paper is entirely concernedwith handling these two difficulties. Surprisingly, we find out in Corol-lary 3.12 that neither difficulty can occur: any minimal clone which hasa nontrivial abelian representation must have a faithful representationas a reduct of a finite cyclic group.Our results enable us to answer a question asked by P. P. P´alfy:Which Mal’cev operations generate minimal clones? Using modularcommutator theory, we show that a minimal clone generated by aMal’cev operation has a nontrivial abelian representation. The answerto P´alfy’s question follows immediately from the classification theorem.(In fact, one could arrange a shorter proof of the answer to P´alfy’s ques-tion by just combining Theorem 3.13 and Lemma 3.9 together with afew arguments from the end of the proof of Lemma 3.4 to make theconnection.) Previously, in [8],´A. Szendrei answered P´alfy’s questionfor minimal clones assumed to have a finite faithful representation. Oursolution does not require a finiteness assumption.2. Minimal ClonesSince a minimal clone is generated by any of its operations which isdifferent from a projection, any minimal clone is unary or idempotent.The idempotent minimal clones may be grouped together according tothe least arity of an operation which is not a projection. If one skipsover the binary idempotent operations which generate minimal clones,MINIMAL CLONES WITH ABELIAN REPRESENTATIONS 3one finds that only very special idempotent operations of high arity cangenerate minimal clones (since every specialization to fewer variablesresults in a projection). We introduce the following terminology for(some) idempotent operations of arity > 2.A Mal’cev operation is a ternary operation p(x, y, z) such that theequationsp(x, y, y) = p(y, y, x) = xhold. A majority operation is a ternary operation M(x, y, z) such thatthe equationsM(x, x, y) = M(x, y, x) = M(y, x, x) = xhold. A minority operation is a ternary operation m(x, y, z) such thatthe equationsm(x, y, y) = m(y, x, y) = m(y, y, x) = xhold. A Pixley operation (sometimes called a 2/3–minority operation)is a ternary operation P (x, y, z) such that the equationsP (x, y, y) = P (x, y, x) = P (y, y, x) = xhold. An (i–th variable) semiprojection is an operation s(x1, . . . , xk) ofarity ≥ 3 which is not a projection, but whenever two arguments areequal then the value of s(x1, . . . , xk) is xi.The following theorem describes the five classes of minimal clones.A proof can be found in Chapter 1 of [7].THEOREM 2.1. A minimal clone is generated by one of the followingtypes of operations:(I) a unary operation not equal to a projection;(II) an idempotent binary operation not equal to a projection;(III) a majority operation;(IV) a minority operation; or(V) a semiprojection. 2The following is an easy exercise.THEOREM 2.2. A clone of class (I) is minimal iff it is generated bya unary operation f which is different from the unary projection andfor which either f(f(x)) = f(x) holds or else fp(x) = x for some primep. 2Note that if P (x, y, z) is a Pixley operation, thenm(x, y, z) := P (P (x, z, y), y, P (y, x, z))is a minority operation. Moreover, this minority operation generatesa proper subclone of the clone generated by P. To see this, let C4 KEITH A. KEARNESbe the clone generated by m(x, y, z). If P (x, y, z) ∈ C,