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 Mat. Zametki, 1973, Volume 14, Issue 5, Pages 703–712 (Mi mz9955)

Nilpotent shifts on manifolds

I. I. Mel'nik

Saratov State University

Abstract: On the lattice of manifolds of all algebras $L$ we study the operator of nilpotent closure $J:\alpha\to\alpha+\mathfrak{R}$, where $\mathfrak{R}$ is a nilpotent manifold of $\Omega$-algebras. With a given system of identities $\Sigma$ defining $\alpha$, we construct a system $\Sigma^*$, giving the manifold $\alpha+\mathfrak{R}$. It is proved that if $\alpha$ does not contain $\mathfrak{R}$, then the lattice of submanifolds of $\alpha+\mathfrak{R}$ is the double of the lattice of submanifolds of $\alpha$. We describe the free and subdirect indecomposable manifolds of algebras $\alpha+\mathfrak{R}$. Let $B\in\alpha+\mathfrak{R}$ and $A$ be a dense retract of $B$. We denote by $\theta(B)$ the lattice of congruences on $B$. The theorem is proved: $\theta(B)$ is a complemented lattice if and only if $\theta(A)$ is a complemented lattice.

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English version:
Mathematical Notes, 1973, 14:5, 962–966

Bibliographic databases:

UDC: 512

Citation: I. I. Mel'nik, “Nilpotent shifts on manifolds”, Mat. Zametki, 14:5 (1973), 703–712; Math. Notes, 14:5 (1973), 962–966

Citation in format AMSBIB
\Bibitem{Mel73} \by I.~I.~Mel'nik \paper Nilpotent shifts on manifolds \jour Mat. Zametki \yr 1973 \vol 14 \issue 5 \pages 703--712 \mathnet{http://mi.mathnet.ru/mz9955} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=366782} \zmath{https://zbmath.org/?q=an:0285.08002} \transl \jour Math. Notes \yr 1973 \vol 14 \issue 5 \pages 962--966 \crossref{https://doi.org/10.1007/BF01462258} 

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This publication is cited in the following articles:
1. Davidova D.S. Movsisyan Yu.M., “Hyperidentities of Weakly Idempotent Lattices”, J. Contemp. Math. Anal.-Armen. Aca., 50:6 (2015), 259–264
2. Movsisyan Yu., Davidova D., “a Complete Characterization of Hyperidentities of the Variety of Weakly Idempotent Lattices”, Tenth International Conference on Computer Science and Information Technologies Revised Selected Papers Csit-2015, ed. Shoukourian S., IEEE, 2015, 41–43
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