Lorenzo Molena,
Università di Verona
Arrow-determined
congruences and forcing ideals of Hasse quivers
Abstract:
We give a brief summary of the necessary prerequisites of
lattice theory, in particular defining lattice congruences and
recalling their properties.
Next we define the hasse quiver of a lattice L and describe
a natural lattice structure on the set of its forcing ideals,
ideal (Hasse L).
Successively, we introduce some interesting kinds of lattices
and lattice congruences, namely the weakly atomic lattices, the
completely semidistributive bialgebraic lattices, and the
arrow-determined congruence relations.
Finally we prove that a complete congruence Θ on a complete
lattice L is arrow determined iff L/Θ is weakly atomic, and
we give a sketch of the proof of the isomorphism Conᶜᵃ L
≅ ideal(Hasse L),
where Conᶜᵃ L is the lattice of complete arrow-determined
congruence relations, and L is a completely semidistributive
lattice.
Reference:
Laurent Demonet, Osamu Iyama, Nathan Reading, Idun Reiten, and
Hugh Thomas. Lattice theory of torsion classes: Beyond
τ-tilting theory. Transactions of the American Mathematical
Society, Series B, 2023, 542–612
.