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 category 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 (finitary) algebras in the sense of Birkhoff's paper, but also to algebras that satisfy certain equational implications (such as cancellative semigroups).

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