Axiomatizable class of algebraic structures which is not the reduct of a variety

Question

In the signature, (+), the class of groups is an axiomatizable class of algebraic structures which, though not a variety in that particular signature, is the reduct of a variety in the signature (+, 0, -). But does there exist an axiomatizable class of algebraic structures, maybe even finitely axiomatizable, which is not even the reduct of a variety? By algebraic structure, I mean one without relation symbols, only functions and constants.

Answer 1

The class of fields has this property, since the product of two fields is not a field.

Or, take the class of algebras (in a given language) of size exactly thirteen (this is indeed axiomatizable by a single first-order sentence).