分配格和素域生成的半环簇

The Variety of Semirings Generated by Distributive Lattices and Prime Fields

中文关键词:  素域  分配格  次直不可约半环  簇.

英文关键词:prime field  distributive lattice  subdirectly irreducible semiring  variety.

中文摘要:

      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.

英文摘要:

      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.