The Variety of Semirings Generated by Distributive Lattices and Prime Fields

DOI：10.3770/j.issn:2095-2651.2014.05.003

 作者 单位 邵勇 西北大学数学学院, 陕西 西安 710127 任苗苗 西北大学数学学院, 陕西 西安 710127

令${\cal V}$ 表示由而元素分配格 $B_2$ 和$k$个素域$F_{p_{1}},\,\cdots,\,F_{p_{k}}$所生成的半环簇,即${\cal V}={\bf HSP}\{B_{2},\,F_{p_{1}},\,\cdots,\,F_{p_{k}}\}$. 本文证明了半环簇${\cal V}$是有限基底的. 进一步, 在同构意义下得到了二元素分配格 $B_{2}$ 和素域$F_{p_{1}},\,\cdots,\,F_{p_{k}}$是${\cal V}$中仅有的次直不可约成员.所获结果推广了已有的相关结果.

Let ${\cal V}$ be the variety generated by two-element distributive lattice $B_2$ and $k$ prime fields $F_{p_{1}},\ldots,F_{p_{k}}$. That is to say that ${\cal V}={\bf HSP}\{B_{2},\,F_{p_{1}},\ldots,F_{p_{k}}\}$. It is proved that the variety ${\cal V}$ is finitely based. Also, the two-element distributive lattice $B_{2}$ and prime fields $F_{p_{1}},\ldots,F_{p_{k}}$ are, up to isomorphism, the only subdirectly irreducible semirings in ${\cal V}$. Some known results are extended and enriched.