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Congruence properties of algebras in nilpotent shifts of varieties. (English) Zbl 0821.08009
Denecke, K. (ed.) et al., General algebra and discrete mathematics. Proceedings of the 4th conference on discrete mathematics, Potsdam, Germany, 1993. Lemgo: Heldermann Verlag. Res. Expo. Math. 21, 35-46 (1995).
The nilpotent shift \(N(V)\) of a variety \(V\) is the variety defined by those identities \(t_ 1= t_ 2\) of \(V\) where neither of the terms \(t_ 1\), \(t_ 2\) is a variable [cf. I. I. Mel’nik, Mat. Zametki 14, 703-712 (1973; Zbl 0285.08002)]. Here is a sample of the theorems proved in the present paper: If \(V\neq N(V)\), then the subdirectly irreducible algebras of \(N(V)\) are those of \(V\) and the two-element constant algebra. If \(V\) is congruence permutable, then an algebra in \(N(V)\) has permutable congruences if and only if it belongs to \(V\) or has exactly two elements.
For the entire collection see [Zbl 0810.00008].
Reviewer: M.Armbrust (Köln)

MSC:
08B99 Varieties
08A30 Subalgebras, congruence relations
08B10 Congruence modularity, congruence distributivity
08B26 Subdirect products and subdirect irreducibility
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