Term functions of semigroups

Given a semigroup S, each word in k variables induces a k-ary term function on S. By a classical result from clone theory, term functions can also be described implicitly: On a finite S, the term functions are exactly those functions that preserve every subsemigroup of every finite power of S. Note that this set of all subsemigroups of finite powers (i.e., relations on S) is always infinite. I will consider the following question: Given a finite semigroup S, is there a finite set of relations that determines the term functions on S? The answer is yes for almost all finite semigroups, in particular, for nilpotent semigroups, commutative semigroups (Davey, Jackson, Pitkethly, Szabo, 2011), and groups (Aichinger, Mayr, McKenzie, 2009). So far no semigroup is known for which the answer is no. I will discuss some proofs and connections with the question whether a semigroup is finitely based.