We determine the possible sizes of algebras of finite type that are nilpotent, nonabelian, dualizable, and generate a congruence modular variety. We also show that localization preserves dualizability, then use this result to construct a finite nilpotent algebra A that is nondualizable, yet such that no algebra in ISP(A) has a supernilpotent nonabelian congruence. Finally, we show that if A is a Mal’cev algebra, then higher commutators in any localization of A are closely controlled by higher commutators in subalgebras of A.
Nilpotence and Dualizability of Algebras of Finite Type