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arXiv · 1612.07648

The lattice of congruence lattices of algebras on a finite set

Danica Jakubíková-Studenovská·Reinhard Pöschel·Sándor Radeleczki
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Abstract

The congruence lattices of all algebras defined on a fixed finite set $A$ ordered by inclusion form a finite atomistic lattice $\mathcal E$. We describe the atoms and coatoms. Each meet-irreducible element of $\mathcal E$ being determined by a single unary mapping on $A$, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set $A$. Using these characterizations we deduce several properties of the lattice $\mathcal E$; in particular, we prove that $\mathcal E$ is tolerance-simple whenever $|A|\geq 4$.

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Danica Jakubíková-Studenovská, Reinhard Pöschel, Sándor Radeleczki. 2016-12-21. The lattice of congruence lattices of algebras on a finite set. https://arxiv.org/abs/1612.07648

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