The Semiring Variety Generated by Any Finite Number of Finite Fields and Distributive Lattices

Yong Shao, Siniša Crvenković, Melanija Mitrović

We study the semiring variety $\mathbf{V}$ generated by any finite number of finite fields $F_1,\dots,F_k$ and two-element distributive lattice $B_2$, i.e., $\mathbf{V}=\operatorname{HSP}\{B_2,F_1,\dots,F_k\}$. It is proved that $\mathbf{V}$ is hereditarily finitely based, and that, up to isomorphism, $B_2$ and all subfields of $F_1,\dots,F_k$ are the only subdirectly irreducible semirings in $\mathbf{V}$.