Suppose $\mathcal{C}$ is a variety in the sense of universal algebra with the additional stipulation that all algebras within $\mathcal{C}$ are finite. Then we'll say direct products are cancellative for $\mathcal{C}$ if for any $H,J, K \in \mathcal{C}$ we have $H \times K \cong J \times K \Rightarrow H \cong J$.
It's well known that direct products are cancellative for finite groups, however, they're not cancellative for finite quasigroups. Is it known what conditions make direct products necessarily cancellative for a class of finite algebras as described above? Or does anyone know where I could find more information on this subject?
There are results of this type summarized in Section 5.7 of
Algebras, Lattices, Varieties, volume 1
R. Mckenzie, G. McNulty, W. Taylor
Wadsworth & Brooks/Cole Mathematics Series, 1987.
For example, Corollary 3 (due to L. Lovasz, 1967) to Theorem 5.23 states that
If $A, B, C$ are finite algebras of the same type and $C$ has a 1-element subuniverse, then $A\times C\cong B\times C$ implies $A\cong B$.