Joins of subalgebras and normals in 0-regular varieties

Abstract

In any 0-normal variety (0-regular variety in which {0} is a subalgebra), every congruence class containing 0 is a subalgebra. These “normal subalgebras” of a fixed algebra constitute a lattice, isomorphic to its congruence lattice. We are interested in those 0-normal varieties for which the join of two normal subalgebras in the lattice of normal subalgebras of an algebra equals their join in the lattice of subalgebras, as happens with groups and rings. We characterise this property in terms of a Mal’cev condition, and use examples to show it is strictly stronger than being ideal determined but strictly weaker than being 0-coherent (classically ideal determined) and does not imply congruence permutability.