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Relations of imbeddability and epimorphism on congruence-distributive varieties. (English. Russian original) Zbl 0619.08005
Algebra Logic 24, 388-401 (1985); translation from Algebra Logika 24, No. 5, 588-607 (1985).
For any class K of universal algebras IK denotes the collection of the isomorphism types of K-algebras, $$<IK;\leq >$$ $$(<IK=;\ll >)$$ denotes the quasiordered class such that $$a\leq b$$ iff for the algebras $${\mathfrak A},{\mathfrak B}\in K$$ the isomorphism types of $${\mathfrak A}$$, $${\mathfrak B}$$ are a, b, respectively, $${\mathfrak A}$$ is isomorphic to some subalgebra of $${\mathfrak B}$$ ($${\mathfrak A}$$ is a homomorphic image of $${\mathfrak B})$$. In this paper $$<IK;\leq >$$ and $$<IK;\ll >$$ are studied in the context of a natural extension of the problem of the thin spectrum for congruence-distributive varieties K. In particular for such varieties it is proved the following: 1) Any countable quasiordered set is isomorphically imbeddable in $$<IK_{\aleph_ 1};\ll >$$ $$(K_{\kappa}$$ is the class of K-algebras of power $$\kappa)$$; 2) Under the assumption of Martin’s axiom, if K is a variety with extendable congruences then for any natural number n, any n- element set A and any quasiorders $$\leq_ 1$$, and $$\leq_ 2$$ on A there exists an isomorphic imbedding of $$<A;\leq_ 1,\leq_ 2>$$ in $$<IK;\leq,\ll >$$.
Reviewer: S.R.Kogalovskij

##### MSC:
 08B10 Congruence modularity, congruence distributivity 08A30 Subalgebras, congruence relations 03C05 Equational classes, universal algebra in model theory
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##### References:
 [1] A. G. Pinus, ”On the Cartesian product operation,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 8, 51–53 (1983). · Zbl 0539.08004 [2] J. Adamek, V. Koubek, and N. Trnkova, ”Sums of Boolean spaces represent every group,” Pac. J. Math.,61, No. 1, 1–6 (1975). · Zbl 0338.20062 [3] W. Hanf, ”On some fundamental problems concerning isomorphism of Boolean algebras,” Math. Scand.,5, No. 2, 205–217 (1957). · Zbl 0081.26101 [4] A. L. S. Corner, ”On a conjecture of Pierce concerning direct decompositions of Abelian groups,” in: Proc. Colloq. Abelian Groups, Tihany, September, 1963, L. Fuchs and E. T. Schmidt (eds.), Hungarian Academy of Sciences, Budapest (1964), pp. 43–48. [5] B. Jonsson, ”On isomorphism types of groups and other algebraic systems,” Math. Scand.,5, 224–229 (1957). · Zbl 0081.26201 [6] S. Koppelberg, ”A lattice structure on the isomorphism types of complete Boolean algebras,” in: Set Theory and Model Theory, Lecture Notes in Math., No. 872, Springer-Verlag, Berlin (1981), pp. 98–126. [7] J. Ketonen, ”The structure of countable Boolean algebras,” Ann. Math.,108, No. 1, 41–89 (1978). · Zbl 0418.06006 [8] A. G. Pinus, Elementary Theories of Semigroup Operations on the Class of Linear Orders, Manuscript deposited at VINITI, No. 4161-82 (1982). [9] A. G. Pinus, ”On the relations ’epimorphism’ and ’imbeddability’ on linear orders,” in: Ordered Sets and Lattices [in Russian], No. 8, Saratov (1982), pp. 81–91. [10] A. Levy, Basic Set Theory, Springer-Verlag, Berlin (1979). · Zbl 0404.04001 [11] R. Magari, ”Una dimostrazione del fatto che ogni varieta ammette algebre semplici,” Ann. Univ. Ferrara, Sez. VII,14, No. 1, 1–4 (1969). · Zbl 0247.08016 [12] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York (1981). · Zbl 0478.08001 [13] A. G. Pinus, ”The spectrum of rigid systems of Horn classes,” Sib. Mat. Zh.,22, No. 5, 153–157 (1981). · Zbl 0481.03018 [14] S. Burris, ”Boolean powers,” Algebra Universalis,5, No. 3, 341–360 (1975). · Zbl 0328.08003 [15] K. Kunen and F. D. Tall, ”Between Martin’s axiom and Souslin’s hypothesis,” Fund. Math.,102, 173–181 (1979). · Zbl 0415.03040 [16] M. Weese, ”Mad families and ultrafilters,” Proc. Am. Math. Soc.,80, No. 3, 475–477 (1980). · Zbl 0473.03042 [17] W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer-Verlag, New York (1974).
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