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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

08B10 Congruence modularity, congruence distributivity
08A30 Subalgebras, congruence relations
03C05 Equational classes, universal algebra in model theory
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