The flat semirings with nilpotent multiplicative reducts

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