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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 the term condition in this paper we shall underline the positions where the arguments change in order to make the application more transparent Our project is to describe the minimal clones which have nontrivial abelian representations We approach minimal clones by looking at their nontrivial abelian representations because of an interesting fact about nonunary minimal clones It may be stated as follows nontrivial abelian representations must be faithful The classification problem for minimal clones with a nontrivial abelian representation reduces to the classification problem for minimal clones of abelian algebras and this is a problem easily solved The description of the minimal subclones of a module follows immediately from our main theorem which is Theorem 3 11 However our result is more than this In the first place in our arguments we must allow the possibility that some abelian representations are unrelated to modules But even in the case where a clone C has a nontrivial representation as a reduct of a module we must worry about what happens if this representation is not faithful This paper is entirely concerned with handling these two difficulties Surprisingly we find out in Corollary 3 12 that neither difficulty can occur any minimal clone which has a nontrivial abelian representation must have a faithful representation as a reduct of a finite cyclic group Our results enable us to answer a question asked by P P Pa lfy Which Mal cev operations generate minimal clones Using modular commutator theory we show that a minimal clone generated by a Mal cev operation has a nontrivial abelian representation The answer to Pa lfy s question follows immediately from the classification theorem In fact one could arrange a shorter proof of the answer to Pa lfy s question by just combining Theorem 3 13 and Lemma 3 9 together with a few arguments from the end of the proof of Lemma 3 4 to make the connection Previously in 8 A Szendrei answered Pa lfy s question for minimal clones assumed to have a finite faithful representation Our solution does not require a finiteness assumption 2 Minimal Clones Since a minimal clone is generated by any of its operations which is different from a projection any minimal clone is unary or idempotent The idempotent minimal clones may be grouped together according to the least arity of an operation which is not a projection If one skips over the binary idempotent operations which generate minimal clones MINIMAL CLONES WITH ABELIAN REPRESENTATIONS 3 one finds that only very special idempotent operations of high arity can generate minimal clones since every specialization to fewer variables results 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 the equations p x y y p y y x x hold A majority operation is a ternary operation M x y z such that the equations M x x y M x y x M y x x x hold A minority operation is a ternary operation m x y z such that the equations m x y y m y x y m y y x x hold A Pixley operation sometimes called a 2 3 minority operation is a ternary operation P x y z such that the equations P x y y P x y x P y y x x hold An i th variable semiprojection is an operation s x1 xk of arity 3 which is not a projection but whenever two arguments are equal 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 following types 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 2 The following is an easy exercise THEOREM 2 2 A clone of class I is minimal iff it is generated by a unary operation f which is different from the unary projection and for which either f f x f x holds or else f p x x for some prime p 2 Note that if P x y z is a Pixley operation then m x y z P P x z y y P y x z is a minority operation Moreover this minority operation generates a proper subclone of the clone generated by P To see this let C 4 KEITH A KEARNES be the clone generated by m x y z If …


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