878 research outputs found
The possible values of critical points between strongly congruence-proper varieties of algebras
We denote by Conc(A) the semilattice of all finitely generated congruences of
an (universal) algebra A, and we define Conc(V) as the class of all isomorphic
copies of all Conc(A), for A in V, for any variety V of algebras. Let V and W
be locally finite varieties of algebras such that for each finite algebra A in
V there are, up to isomorphism, only finitely many B in W such that A and B
have isomorphic congruence lattices, and every such B is finite. If Conc(V) is
not contained in Conc(W), then there exists a semilattice of cardinality aleph
2 in Conc(V)-Conc(W). Our result extends to quasivarieties of first-order
structures, with finitely many relation symbols, and relative congruence
lattices. In particular, if W is a finitely generated variety of algebras, then
this occurs in case W omits the tame congruence theory types 1 and 5; which, in
turn, occurs in case W satisfies a nontrivial congruence identity. The bound
aleph 2 is sharp
The possible values of critical points between varieties of lattices
We denote by Conc(L) the semilattice of all finitely generated congruences of
a lattice L. For varieties (i.e., equational classes) V and W of lattices such
that V is contained neither in W nor its dual, and such that every simple
member of W contains a prime interval, we prove that there exists a bounded
lattice A in V with at most aleph 2 elements such that Conc(A) is not
isomorphic to Conc(B) for any B in W. The bound aleph 2 is optimal. As a
corollary of our results, there are continuum many congruence classes of
locally finite varieties of (bounded) modular lattices
Tame stacks and log flat torsors
We compare tame actions in the category of schemes with torsors in the
category of log schemes endowed with the log flat topology. We prove that
actions underlying log flat torsors are tame. Conversely, starting from a tame
cover of a regular scheme that is a fppf torsor on the complement of a divisor
with normal crossings, it is possible to build a log flat torsor that dominates
this cover. In brief, the theory of log flat torsors gives a canonical approach
to the problem of extending torsors into tame covers.Comment: 17 pages, LaTe
Critical points between varieties generated by subspace lattices of vector spaces
We denote by Conc(A) the semilattice of compact congruences of an algebra A.
Given a variety V of algebras, we denote by Conc(V) the class of all
semilattices isomorphic to Conc(A) for some A in V. Given varieties V1 and V2
varieties of algebras, the critical point of V1 under V2, denote by crit(V1;V2)
is the smalest cardinality of a semilattice in Conc(V1) but not in Conc(V2).
Given a finitely generated variety V of modular lattices, we obtain an integer
l, depending of V, such that crit(V;Var(Sub F^n)) is at least aleph_2 for any n
> 1 and any field F. In a second part, we prove that crit(Var(Mn);Var(Sub
F^3))=aleph_2, for any finite field F and any integer n such that 1+card F< n.
Similarly crit(Var(Sub F^3);Var(Sub K^3))=aleph_2, for all finite fields F and
K such that card F>card K
- …