Infinite combinatorial issues raised by lifting problems in universal algebra

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (k,r,l){\to}m (for proving that a critical point is small). In the middle, we find l-lifters of posets and the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea

Lifting retracted diagrams with respect to projectable functors

We prove a general categorical theorem that enables us to state that under certain conditions, the range of a functor is large. As an application, we prove various results of which the following is a prototype: If every diagram, indexed by a lattice, of finite Boolean (v,0)-semilattices with (v,0)-embeddings, can be lifted with respect to the \Conc functor on lattices, then so can every diagram, indexed by a lattice, of finite distributive (v,0)-semilattices with (v,0-embeddings. If the premise of this statement held, this would solve in turn the (still open) problem whether every distributive algebraic lattice is isomorphic to the congruence lattice of a lattice. We also outline potential applications of the method to other functors, such as the $R\mapsto V(R)$ functor on von Neumann regular rings

Uniform interpolation and coherence

A variety V is said to be coherent if any finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that V is coherent if and only if it satisfies a restricted form of uniform deductive interpolation: that is, any compact congruence on a finitely generated free algebra of V restricted to a free algebra over a subset of the generators is again compact. A general criterion is obtained for establishing failures of coherence, and hence also of uniform deductive interpolation. This criterion is then used in conjunction with properties of canonical extensions to prove that coherence and uniform deductive interpolation fail for certain varieties of Boolean algebras with operators (in particular, algebras of modal logic K and its standard non-transitive extensions), double-Heyting algebras, residuated lattices, and lattices

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

Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction

A (v,0)-semilattice is ultraboolean, if it is a directed union of finite Boolean (v,0)-semilattices. We prove that every distributive (v,0)-semilattice is a retract of some ultraboolean (v,0)-semilattices. This is established by proving that every finite distributive (v,0)-semilattice is a retract of some finite Boolean (v,0)-semilattice, and this in a functorial way. This result is, in turn, obtained as a particular case of a category-theoretical result that gives sufficient conditions, for a functor $Pi$, to admit a right inverse. The particular functor $Pi$ used for the abovementioned result about ultraboolean semilattices has neither a right nor a left adjoint