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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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