Congruence lattices of lattices
The
"Congruence Lattice Problem" in the Wikipedia
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Friedrich Wehrung:
There
exists a distributive algebraic lattice which is not isomorphic to the
congruence lattice of any lattice.
This
lattice has א ω+1 compact elements.
Pavel Růžička:
There is a distributive
join semilattice S of size
א i2
that is not isomorphic to the congruence lattice
of compact congruences of any lattice.
Růžička, Tůma and Wehrung:
There is a distributive
algebraic lattice that is not isomorphic to the congruence lattice
of any algebra with permutable congruences.
Milan Ploscica
Local separation in
distributive semilattices, AU 54, 323-335
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Papers:
Slide show gallery
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Open
problems
Problem 1.
Does there
exists a distributive algebraic lattice which is not isomorphic to the
congruence lattice of any algebraic structure with only finitely many operations ?
What lattices are congruence lattices of algebras with only finitely many
operations?
Problem 2.
( Lampe) Is there an algebra A with exactly
אi0
operations such that there is no B with only finitely many operations and
with Con(B) isomorphic to Con(A) ?
Problem 3.
Is every finite distributive lattice isomorphic to the congruence lattice of
an isoform modular lattice ?
Problem 4.
(Lampe) Which algebraic lattices are isomorphic to the congruence lattices
of algebras having one element subalgebras ?