Varieties having directly decomposable congruence classes
Varieties having distributive lattices of quasiorders
Varieties satisfying ideal equalities
Varieties satisfying the triangular scheme need not be congruence distributive
A diagrammatic scheme characterizing congruence distributivity of congruence permutable algebras was introduced by the first author in 2001. It is known under the name Triangular Scheme. It is known that every congruence distributive algebra satisfies this scheme and an algebra satisfying the Triangular Scheme which is not congruence distributive was found by E. K. Horváth, G. Czédli and the autor in 2003. On the other hand, it was an open problem if a variety of algebras satisfying the Triangular...
Varieties with modular and distributive lattices of symmetric or reflexive relations
Varieties with polynomially many models, I
A characterization of locally finite congruence modular varieties with the number of at most k-generated models being bounded from above by a polynomial in k is given. These are exactly the varieties polynomially equivalent to the varieties of unitary modules over a finite ring of finite representation type.
Varieties with tolerance and congruence extension property