Nilpotent shifts on manifolds

I. I. Mel'nik

Matematičeskie zametki, Tome 14 (1973) no. 5, pp. 703-712.

Résumé

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.