Essentially Minimal Clones of Rank 3 on a Three-Element Set

For a non-empty set A, a clone C on A is essentially minimal if C is not a minimal clone but is minimal among all clones on A containing essential functions. For a finite set A, the rank of an essentially minimal clone C on A, which is the least arity of generators of C, is no greater than |A|. We determined, in 2013, all essentially minimal clones of rank 2 on the three-element set E₃. In this paper, we continue and determine all essentially minimal clones of rank 3 on E₃. There are 12 essentially minimal clones of rank 3, which are divided into two conjugate classes. This completes the search of essentially minimal clones on E₃.