Uncountable critical points for congruence lattices

Abstract

The critical point between two classes ${{\mathcal K}}$ and ${{\mathcal L}}$ of algebras is the cardinality of the smallest semilattice isomorphic to the semilattice of compact congruences of some algebra in ${{\mathcal K}}$, but not in ${{\mathcal L}}$. Our paper is devoted to the problem of determining the critical point between two finitely generated congruence-distributive varieties. For a homomorphism ${\varphi: S \rightarrow T}$ of ${(0, \vee)}$-semilattices and an automorphism ${\tau}$ of T, we introduce the concept of a ${\tau}$-symmetric lifting of ${\varphi}$. We use it to prove a criterion which ensures that the critical point between two finitely generated congruence-distributive varieties is less or equal to ${\aleph_{1}}$. We illustrate the criterion by constructing two new examples with the critical point exactly ${\aleph_{1}}$.