Categories of partial algebras for critical points between varieties of algebras

Abstract

We denote by Conc A the $${(\vee, 0)}$$ -semilattice of all finitely generated congruences of an algebra A. A lifting of a $${(\vee, 0)}$$ -semilattice S is an algebra A such that $${S \cong {\rm Con}_{\rm c} A}$$ . The assignment Conc can be extended to a functor. The notion of lifting is generalized to diagrams of $${(\vee, 0)}$$ -semilattices. A gamp is a partial algebra endowed with a partial subalgebra together with a semilattice-valued distance; gamps form a category that lends itself to a universal algebraic-type study. The raison d’être of gamps is that any algebra can be approximated by its finite subgamps, even in case it is not locally finite. Let $${\mathcal{V}}$$ and $${\mathcal{W}}$$ be varieties of algebras (on finite, possibly distinct, similarity types). Let P be a finite lattice. We assume the existence of a combinatorial object, called an $${\aleph_0}$$ -lifter of P, of infinite cardinality $${\lambda}$$ . Let $${\vec{A}}$$ be a P-indexed diagram of finite algebras in $${\mathcal{V}}$$ . If $${{\rm Con}_{\rm c} \circ \vec{A}}$$ has no partial lifting in the category of gamps of $${\mathcal{W}}$$ , then there is an algebra $${A \in \mathcal{V}}$$ of cardinality $${\lambda}$$ such that Conc A is not isomorphic to Conc B for any $${B \in \mathcal{W}}$$ . This makes it possible to generalize several known results. In particular, we prove the following theorem, without assuming that $${\mathcal{W}}$$ is locally finite. Let $${\mathcal{V}}$$ be locally finite and let $${\mathcal{W}}$$ be congruence-proper (i.e., congruence lattices of infinite members of $${\mathcal{W}}$$ are infinite). The following equivalence holds. Every countable $${(\vee, 0)}$$ -semilattice with a lifting in $${\mathcal{V}}$$ has a lifting in $${\mathcal{W}}$$ if and only if every $${\omega}$$ -indexed diagram of finite $${(\vee, 0)}$$ -semilattices with a lifting in $${\mathcal{V}}$$ has a lifting in $${\mathcal{W}}$$ . Gamps are also applied to the study of congruence-preserving extensions. Let $${\mathcal{V}}$$ be a non-semidistributive variety of lattices and let n ≥ 2 be an integer. There is a bounded lattice $${A \in \mathcal{V}}$$ of cardinality $${\aleph_1}$$ with no congruence n-permutable, congruence-preserving extension. The lattice A is constructed as a condensate of a square-indexed diagram of lattices.