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All clones are centralizer clones. (English) Zbl 1201.03017
An abstract clone is a small category whose object set consists of all finite powers of a base object \(a\), with \(a^0\) as terminal object, and in which for every \(n\in\omega\) an \(n\)-tuple of product projections \(\pi^n_0,\dots,\pi^n_{n-1}\) is specified. Similarly to representations of abstract groups as permutation groups acting on a set \(X\), abstract clones can always be represented as concrete clones of finitary functions on a set \(X\), i.e., as sets of finitary functions on \(X\) closed under composition and containing all projections.
In the present paper, it is shown that abstract clones can even be represented as a specific type of concrete clones, namely as centralizer clones of certain algebraic systems; the centralizer clone of an algebraic system is the set of homomorphisms from finite powers of the system into the system. The main theorem states that every abstract clone can be represented as the centralizer clone of an algebraic system which has precisely one unary relation and \(\kappa\) unary operations, where \(\kappa\) is the number of morphisms of the abstract clone. Moreover, if the abstract clone does not contain any virtual constants, i.e., morphisms that “look” constant without being explicitly forced to be constant by factoring through nullary morphisms, then the unary relation can be avoided and the clone can be represented as the centralizer clone of an algebra which has precisely \(\kappa\) unary operations. Finally, if the abstract clone is in addition countable, then two unary operations suffice.

MSC:
03C05 Equational classes, universal algebra in model theory
08A40 Operations and polynomials in algebraic structures, primal algebras
08A60 Unary algebras
18B15 Embedding theorems, universal categories
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