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Register a new userThe possible values of critical points between strongly congruence-proper varieties of algebras
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Authors
Pierre Gillibert
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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.
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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.
For a class V of algebras, denote by Conc(V) the class of all semilattices isomorphic to the semilattice Conc(A) of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1,V2) the smallest cardinality of a semilattice in Conc(V1) which is not in Conc(V2) if it exists, infinity otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties V1 and V2, crit(V1,V2) is either finite, or aleph_n for some natural number n, or infinity. We also find two finitely generated modular lattice varieties V1 and V2 such that crit(V1,V2)=aleph_1, thus answering a question by J. Tuma and F. Wehrung.
We study the class of finite lattices that are isomorphic to the congruence lattices of algebras from a given finitely generated congruence-distributive variety. If this class is as large as allowed by an obvious necessary condition, the variety is called congruence FD-maximal. The main results of this paper characterize some special congruence FD-maximal varieties.
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.
We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural D-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counterexamples to naive generalizations of our results. These examples have been a source of motivation for us.
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