The flat semirings with nilpotent multiplicative reducts

In this paper, we focus on the variety $\mathbf{NF}_3$ generated by all flat semirings with $3$-nilpotent multiplicative reducts. By introducing graph semirings, we characterize all subdirectly irreducible members of $\mathbf{NF}_3$. We prove that the variety $\mathbf{NF}_3$ has uncountably many subvarieties and show that every finitely generated subvariety of $\mathbf{NF}_3$ is a Cross variety. Moreover, we demonstrate that $\mathbf{NF}_3$ has a unique limit subvariety, which is generated by all acyclic graph semirings.