Categories of partial algebras for critical points between varieties of algebras

We denote by Con_c 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 Con_c A is not isomorphic to Con_c 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.