The structure of the lattices of quasivarieties (Q1342130)

For an algebraic system \(A\) and a quasivariety \(\mathcal K\) let \(\text{Con}_{{\mathcal K}} A\) be the lattice of all congruence relations \(\theta\) on \(A\) such that \(A/\theta\in {\mathcal K}\). Define the embedding relation \(\leq\) as follows: \(\theta\leq \theta'\) iff \(A/\theta'\) is embeddable into \(A/\theta\). Let \(\text{Sp}(\text{Con}_{{\mathcal K}} A, \leq)\) be the lattice of algebraic \(\leq\)-closed subsets of \(\text{Con}_{{\mathcal K}} A\). The author proves that every lattice \(L_ q({\mathcal K})\) of subquasivarieties of \(\mathcal K\) is isomorphic to the inverse limit of lattices \(\text{Sp}(\text{Con}_{{\mathcal K}} G_ i, \leq)\) for some set of finitely presented systems \(G_ i\) of the quasivariety \(\mathcal K\). In particular, the lattice \(L_ q({\mathcal K})\) is residually finite for every locally finite quasivariety \(\mathcal K\) of finite type. Also some new properties of embedding relations on congruence lattices of free systems are investigated.