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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.

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
Full Text: DOI
[1] Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer Academic Publishers (1990) · Zbl 0698.18001
[2] Cohn P.M.: Universal Algebra. Harper and Row, New York (1965)
[3] Goldstern M., Shelah S.: Clones on regular ordinals. Fund. Math. 173, 1–20 (2002) · Zbl 0997.08004
[4] Goralčík P., Koubek V., Sichler J.: Universal varieties of (0, 1)-lattices. Can. J. Math. 42, 470–490 (1990) · Zbl 0709.18003
[5] Grätzer G.: Universal Algebra, Second Edition. Springer-Verlag, New York (1979)
[6] Hall P.: Some word problems. J. London Math. Soc. 33, 482–496 (1958) · Zbl 0198.02902
[7] Koubek V., Sichler J.: Finitely generated varieties of distributive double p-algebras universal modulo a group. Algebra Universalis 51, 35–79 (2004) · Zbl 1092.06009
[8] Lawvere F.W.: Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A., 50, 869–872 (1963) · Zbl 0119.25901
[9] Lawvere, F. W.: Some algebraic problems in the context of functorial semantics of algebraic theories. Lect. Notes. in Math. 61, pp.41–46. Springer-Verlag, Berlin and New York (1968) · Zbl 0204.33802
[10] McKenzie, R., McNulty, G., Taylor, W.: Algebras, Lattices, Varieties, vol. 1. Brooks/Cole, Monterey, California (1987) · Zbl 0611.08001
[11] Pinsker M.: Clones containing all almost n-ary functions. Algebra Universalis 51, 235–255 (2004) · Zbl 1082.08003
[12] Post E.L.: Introduction to a general theory of elementary propositions. Amer. J. Math. 43, 163–185 (1921) · JFM 48.1122.01
[13] Post, E. L.: The Two-valued Iterative Systems of Mathematical Logic. Annals of Mathematics Studies vol. 5, Princeton University Press, Princeton, N. J. (1941) · Zbl 0063.06326
[14] Pultr A., Trnková V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North Holland, Amsterdam (1980) · Zbl 0418.18004
[15] Rosenberg I.G. Über die funktionale Vollständingkeit in den mehrwertigen Logiken. Rozpravy Československé Akad. Věd, Řada Mat. Přírod. Věd 80 (1970)
[16] Rosenberg I.G.: The set of maximal closed classes of operations on an infinite set A has cardinality \({{{2^{2}}^{|A|}}}\) . Arch. Math. (Basel) 27, 561–568 (1976) · Zbl 0345.02010
[17] Sichler J., Trnková V.: Clones in topology and algebra. Acta Math. Univ. Comenianae 66, 243–260 (1997) · Zbl 0970.08004
[18] Sichler, J., Trnková, V.: Essential operations in centralizer clones. Algebra Universalis, to appear · Zbl 1157.08003
[19] Szendrei, Á.: Clones in Universal Algebra. Les Presses de L’Université de Montréal (1986)
[20] Taylor, W.: Abstract Clone Theory. In Proceedings of the NATO Advanced Study Institute Montréal 1991, Kluwer Academic Publishers, pp. 507–530 (1993) · Zbl 0792.08005
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