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.