By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the taking of homomorphic images within the class of magmas, every semigroup is (isomorphic to) a subdirect product of subdirectly irreducible semigroups. What about cancellative semigroups? More precisely:

Q. Is every cancellative semigroup a subdirect product of subdirectly irreducible, cancellative semigroups?

The issue is that if a semigroup $T$ is the homomorphic image of a cancellative semigroup $S$, then $T$ need not be cancellative. However, I am hoping that there is a generalization of Birkhoff's theorem that applies not only to algebras (in the sense of Birkhoff's paper) but also to algebras that satisfy certain equational identities (such as cancellative semigroups).

Am I hoping for too much? Or am I missing something?