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On clones determined by their initial segments. (English) Zbl 1163.08001
Let \(\mathcal A\) and \(\mathcal B\) be abstract clones. We say that they are locally isomorphic if their \(n\)-segments are isomorphic for every \(n\in\omega\). It is proven for bounded finitary similarity type that two algebras of this type have isomorphic polynomial clones if and only if they have locally isomorphic ones. Examples of algebras of the same type such that 5mm
a)
their type is unbounded finitary and their polynomial clones are locally isomorphic but not isomorphic,
b)
they have one binary and countably many unary operations and their term clones are locally isomorphic but not isomorphic,
c)
they have two unary operations and their centralizer clones are locally isomorphic but not isomorphic
are presented. For a class \(\mathcal C\) of clones, properties of their representability as centralizer clones in categories are investigated.

MSC:
08A40 Operations and polynomials in algebraic structures, primal algebras
03C60 Model-theoretic algebra
08B25 Products, amalgamated products, and other kinds of limits and colimits
18B15 Embedding theorems, universal categories
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References:
[1] J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York. Zbl0695.18001 MR1051419 · Zbl 0695.18001
[2] J. T. Baldwin and J. Berman, Elementary classes of varieties, Houston. J. Math. 7 (1981),473-492. Zbl0487.08007 MR658563 · Zbl 0487.08007
[3] P. M. Cohn, Universal Algebra, Harper and Row, New York, 1965. Zbl0141.01002 MR175948 · Zbl 0141.01002
[4] G. Grätzer, Universal Algebra, Second Edition, Springer-Verlag, New York, 1979. MR538623
[5] P. Hall, Some word problems, J. London Math. Soc. 33 (1958), 482-496. Zbl0198.02902 MR102540 · Zbl 0198.02902
[6] F. W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869-872. Zbl0119.25901 MR158921 · Zbl 0119.25901
[7] F. W. Lawvere, Some algebraic pwblems in the context of functorial semantics of algebraic theories, Lect. N. in Math., vol. 61, Springer-Verlag, Berlin and New York, 1968, pp. 41-46. Zbl0204.33802 MR231882 · Zbl 0204.33802
[8] K. D. Magill, The semigroup of endomorphisms of a Boolean ring, Semigroup Forum 4 (1972),411-416. Zbl0224.06007 MR272690 · Zbl 0224.06007
[9] K. D. Magill, A survey of semigroups of continuons selfmaps, Semigroup Forum 11 ( 1975/76), 189-282. Zbl0338.20088 MR393330 · Zbl 0338.20088
[10] C. J. Maxson, On semigroups of Boolean ring endomorphisms, Semigroup Forum 4 (1972), 78-82. Zbl0262.06011 MR297900 · Zbl 0262.06011
[11] R. N. McKenzie, G. F. McNulty and W. F. Taylor, Algebras, Lattices, Varieties, Volume 1, Wadsworth & Brooks/Cole, Monterey, California, 1987. Zbl0611.08001 · Zbl 0611.08001
[12] R. N. McKenzie and C. Tsinakis, On recovering of a bounded distributive lattice from its endomorphism monoid, Houston J. Math 7 (1981), 525-529. Zbl0492.06009 MR658568 · Zbl 0492.06009
[13] E. L. Post, Introduction to a general theory of elementary propositions, Amer. J. Math. 43 (1921), 163-185. MR1506440 JFM48.1122.01 · JFM 48.1122.01
[14] E. L. Post, The Two-valued Iterative Systems of Mathematical Logic, Annals of Mathematics Studies No. 5, Princeton University Press, Princeton, N. J., 1941. Zbl0063.06326 MR4195 · Zbl 0063.06326
[15] B. M. Schein, Ordered sets, semilattices, distributive lattices and Boolean algebras with homomorphic endomorphism semigroups, Fund. Math. 68 (1970), 31-50. Zbl0197.28902 MR272686 · Zbl 0197.28902
[16] J. Sichler and V. Trnková, Clones in topology and algebra, Acta Math. Univ. Comenianae 66 (1997), 243-260. Zbl0970.08004 MR1620417 · Zbl 0970.08004
[17] J. Sichler and V. Trnková, Clone segment independence in topology and algebra, Acta Sci. Math. (Szeged) 68 (2002), 611-672. Zbl1017.54006 MR1954539 · Zbl 1017.54006
[18] Á. Szendrei, Clones in Universal Algebra, Les Presses de L’Université de Montréal, 1986. Zbl0603.08004 MR859550 · Zbl 0603.08004
[19] W. Taylor, The Clone of a Topological Space, Research and Exposition in Mathematics, vol. 13, 1986, Heldermann Verlag, 1986. Zbl0615.54013 MR879120 · Zbl 0615.54013
[20] V. Trnková, Representability and local representability of algebraic theories, Algebra Universalis 29 (1998), 121-144. Zbl0936.18007 MR1636983 · Zbl 0936.18007
[21] V. Trnková and J. Sichler, All clones are centralizer clones, preprint (2007). Zbl1201.03017 MR2551786 · Zbl 1201.03017
[22] J. Sichler, Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2, sichler@cc.umanitoba.ca
[23] V. Trnková, Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, trnkova@karlin.mff.cuni.cz
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