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Varieties whose skeletons are lattices. (English. Russian original) Zbl 0787.08007
Algebra Logic 31, No. 1, 48-53 (1992); translation from Algebra Logika 31, No. 1, 74-82 (1992).
In Algebra Logika 24, No. 5, 588-607 (1985; Zbl 0619.08005), we introduced the notions of epimorphism and embeddability skeletons of a variety of algebras and studied them in more detail for the case of congruence distributive (c.d.) varieties in a series of subsequent publications. In Algebra Logika 27, No. 3, 316-326 (1988; Zbl 0666.08004), we noted that the skeletons in question take the position between “thin” congruence lattices of free algebras, lattices of subalgebras of universal algebras (their skeletons are, respectively, antiisotonical and isotonical images), and “rough” lattices of subvarieties (these lattices are isotonical images of skeletons).
In connection with the above we raise the following question: Is it always the case, and if not then in what cases, are the epimorphism and embeddability skeletons of a variety of algebras lattices in their own right? In the present note we consider this problem for the case of c.d. varieties.
MSC:
08B10 Congruence modularity, congruence distributivity
08A35 Automorphisms and endomorphisms of algebraic structures
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References:
[1] A. G. Pinus, ”Relations of epimorphism and embeddability on congruence-distributive varieties,” Algebra Logika,24, No. 5, 588–607 (1985). · Zbl 0619.08005
[2] A. G. Pinus, ”Congruence-distributive varieties of algebras,” Itogi Nauki i Tekhniki (Algebra, Topology, and Geometry),26, 45–83 (1988). · Zbl 0679.08004
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