Congruence properties of algebras in nilpotent shifts of varieties. (English) Zbl 0821.08009

Chajda, I.

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

Keywords:

normal identity; congruence relation; congruence permutable variety; nilpotent shift; subdirectly irreducible algebras