Sangalli, Arturo A. L. On the structure and representation of clones. (English) Zbl 0636.08003 Algebra Univers. 25, No. 1, 101-106 (1988). Let C be an abstract clone. We define an equivalence relation on C and show that C is isomorphic to the clone \(\hat C\) of all operations on the quotient set \(X_ C\) preserved by a monoid \(M_ C\) of transformations of \(X_ C\). It turns out that \(<X_ C,\hat C>\) is a countably generated free algebra and \(M_ C\) is the monoid generated by those endomorphisms which fix all but one generator. As an application we prove that all endomorphism monoids and automorphism groups of finite powers of algebras arise, up to isomorphism, from unary algebras. Reviewer: A.A.L.Sangalli Cited in 3 Documents MSC: 08A40 Operations and polynomials in algebraic structures, primal algebras 08A35 Automorphisms and endomorphisms of algebraic structures 08A05 Structure theory of algebraic structures 08A60 Unary algebras Keywords:transformation monoid; clone; operations; free algebra; endomorphism monoids; automorphism groups; unary algebras PDFBibTeX XMLCite \textit{A. A. L. Sangalli}, Algebra Univers. 25, No. 1, 101--106 (1988; Zbl 0636.08003) Full Text: DOI References: [1] P. M. Cohn,Universal algebra, Harper and Row, New York, 1965. [2] T.Evans,Some remarks on the general theory of clones, Colloquia Mathematica Societatis J?nos Bolyai 28. Finite algebra and multiple-valued logic. Szeged (Hungary) (1979), 203-244. [3] E. G. Manes,Algebraic theories, Graduate Texts in Mathematics, Springer-Verlag, New York, 1975. · Zbl 0489.18003 [4] A. A. L. Sangalli,A representation of varieties and their morphisms, Algebra Universalis,17 (1983), 120-128. · Zbl 0523.08006 [5] B. M. Schein andV. S. Trohimenko,Algebras of multiplace functions, Semigroup Forum,17 (1979), 1-64. · Zbl 0397.08001 [6] W.Taylor,A question on representing algebraic theories, unpublished manuscript, 1979. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.