TITLE: Algebras with finite essential arity
QUESTION [6 upvotes]: We are talking about algebras in the universal algebraic sense, that is, a set that $A$ is equipped with a set $F$ of finitary operations on $A$.

Definition: An algebra $(A,F)$ is said to have finite essential arity if there exists a number $n \in \mathbb{N}$ such that each of its term functions depends on at most $n$ variables.

I am mostly interested in the case where $A$ is finite. I have read some results about this,  but I am not sure what exactly is known about these kind of algebras. 
I am particularly interested in identities that hold (or don't hold) for algebras of this kind. I know a few conditions on identities that imply finite essential arity, but not so many that are required or even equivalent. Thus, I would be very interested in examples of this kind.
To ask one more specific question that goes in the same direction: Are there (finite) algebras with finite essential arity that are not strongly abelian? What would be an example for that?

REPLY [4 votes]: A finite algebra in a finite language has finite essential arity if and only if it is strongly nilpotent. This is proved in Section 4 of:
Keith A. Kearnes and Emil W. Kiss, Finite algebras of finite complexity. Discrete Math. 207 (1999), 89--135.