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Congruence lattices forcing nilpotency. (English) Zbl 1401.08001

The paper deals with structural properties of an algebra that are forced by the shape of its congruence lattice. More precisely, a lattice \(\mathbb L\) forces a property \(p\) in a class \(K\) of algebras if for each \(A\in K\), whenever \(\mathrm{Con}(A)\cong \mathbb L\) then \(A\) has the property \(p\).
One of the main results main results of the paper is the following theorem:
Theorem. Let \(\mathbb L\) be a lattice of finite height that is the congruence lattice of some algebra in a congruence modular variety. Then the following conditions are equivalent:
(1)
\(\mathbb L\) forces solvability in the class of algebras generating congruence modular varieties.
(2)
Every algebra \(B\) generating a congruence modular variety with \(\mathrm{Con}(B)\cong \mathbb L\) is solvable.
(3)
The two element lattice \(B_2\) is not a homomorphic image of \(\mathbb L\).
Further, necessary and sufficient conditions for nilpotency (supernilpotency) in the class of all finite expanded groups (in the class of all algebras that generate a congruence modular variety) are proved. For the proofs of the theorems, the author first shows several nice results of purely theoretical character, concerning lattices endowed with the binary operation of taking commutators.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
06C10 Semimodular lattices, geometric lattices
08B10 Congruence modularity, congruence distributivity
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