Abstract
We show that any clone without virtual constants is isomorphic to the centralizer clone of a unary universal algebra, and that adding one unary relation to these unary algebras produces algebraic systems representing any clone.
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Adámek, J., Trnková, V.: Automata and Algebras in Categories. Kluwer Academic Publishers (1990)
Cohn P.M.: Universal Algebra. Harper and Row, New York (1965)
Goldstern M., Shelah S.: Clones on regular ordinals. Fund. Math. 173, 1–20 (2002)
Goralčík P., Koubek V., Sichler J.: Universal varieties of (0, 1)-lattices. Can. J. Math. 42, 470–490 (1990)
Grätzer G.: Universal Algebra, Second Edition. Springer-Verlag, New York (1979)
Hall P.: Some word problems. J. London Math. Soc. 33, 482–496 (1958)
Koubek V., Sichler J.: Finitely generated varieties of distributive double p-algebras universal modulo a group. Algebra Universalis 51, 35–79 (2004)
Lawvere F.W.: Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A., 50, 869–872 (1963)
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)
McKenzie, R., McNulty, G., Taylor, W.: Algebras, Lattices, Varieties, vol. 1. Brooks/Cole, Monterey, California (1987)
Pinsker M.: Clones containing all almost n-ary functions. Algebra Universalis 51, 235–255 (2004)
Post E.L.: Introduction to a general theory of elementary propositions. Amer. J. Math. 43, 163–185 (1921)
Post, E. L.: The Two-valued Iterative Systems of Mathematical Logic. Annals of Mathematics Studies vol. 5, Princeton University Press, Princeton, N. J. (1941)
Pultr A., Trnková V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North Holland, Amsterdam (1980)
Rosenberg I.G. Über die funktionale Vollständingkeit in den mehrwertigen Logiken. Rozpravy Československé Akad. Věd, Řada Mat. Přírod. Věd 80 (1970)
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)
Sichler J., Trnková V.: Clones in topology and algebra. Acta Math. Univ. Comenianae 66, 243–260 (1997)
Sichler, J., Trnková, V.: Essential operations in centralizer clones. Algebra Universalis, to appear
Szendrei, Á.: Clones in Universal Algebra. Les Presses de L’Université de Montréal (1986)
Taylor, W.: Abstract Clone Theory. In Proceedings of the NATO Advanced Study Institute Montréal 1991, Kluwer Academic Publishers, pp. 507–530 (1993)
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Presented by A. Szendrei.
To Václav Koubek on his 60th birthday
The first author gratefully acknowledges the support of MSM 0021620839, a project of the Czech Ministry of Education, and of the grant 201/06/0664 by the Grant Agency of Czech Republic. The second author gratefully acknowledges the support provided by the NSERC of Canada.
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Trnková, V., Sichler, J. All clones are centralizer clones. Algebra Univers. 61, 77 (2009). https://doi.org/10.1007/s00012-009-0004-4
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DOI: https://doi.org/10.1007/s00012-009-0004-4