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Quotients of Strongly Proper Forcings and Guessing Models
We prove that a wide class of strongly proper forcing posets have quotients
with strong properties. Specifically, we prove that quotients of forcing posets
which have simple universal strongly generic conditions on a stationary set of
models by certain nice regular suborders satisfy the -approximation
property. We prove that the existence of stationarily many -guessing
models in , for sufficiently large cardinals ,
is consistent with the continuum being arbitrarily large, solving a problem of
Viale and Weiss
Lattice theory of torsion classes
The aim of this paper is to establish a lattice theoretical framework to
study the partially ordered set of torsion
classes over a finite-dimensional algebra . We show that
is a complete lattice which enjoys very strong
properties, as bialgebraicity and complete semidistributivity. Thus its Hasse
quiver carries the important part of its structure, and we introduce the brick
labelling of its Hasse quiver and use it to study lattice congruences of
. In particular, we give a
representation-theoretical interpretation of the so-called forcing order, and
we prove that is completely congruence
uniform. When is a two-sided ideal of , is a lattice quotient of which is
called an algebraic quotient, and the corresponding lattice congruence is
called an algebraic congruence. The second part of this paper consists in
studying algebraic congruences. We characterize the arrows of the Hasse quiver
of that are contracted by an algebraic
congruence in terms of the brick labelling. In the third part, we study in
detail the case of preprojective algebras , for which
is the Weyl group endowed with the weak
order. In particular, we give a new, more representation theoretical proof of
the isomorphism between and the Cambrian
lattice when is a Dynkin quiver. We also prove that, in type , the
algebraic quotients of are exactly its
Hasse-regular lattice quotients.Comment: 65 pages. Many improvements compared to the first version (in
particular, more discussion about complete congruence uniform lattices
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