A finitely based finite semiring generates a variety with continuum many subvarieties

This paper establishes the existence of a finitely based finite semiring whose variety contains a continuum of subvarieties; such a variety is said to be of type $2^{{\mathrm{\aleph }}_0}$. Using the homomorphism theory of Kneser graphs, we prove that the 3-element semiring $S_{53}$ is the first known example with this property. Moreover, $S_{53}$ belongs to the variety of the max-plus semiring $(\mathbb{N},\max ,+)$, which therefore is also of type $2^{{\mathrm{\aleph }}_0}$. For the finitely based 4-element semiring $B_0$, we demonstrate that its variety contains infinitely many subvarieties and suggest that $B_0$ could be another potential example of type $2^{{\mathrm{\aleph }}_0}$.