含幺元的半格和群的Zappa-Sz\'{e}p积上的同余

Congruences on Zappa-Sz\'{e}p Products of Semilattices with An Identity and Groups

中文关键词:  Zappa-Sz\'{e}p积  同余  同余对.

英文关键词:Zappa-Sz\'{e}p product  congruence  congruence pairs.

中文摘要:

      设$P=E\bowtie G$是含幺元半格$E$和群$G$的 Zappa-Sz\'{e}p积. 首先, 我们给出P上同余对的概念. 然后,我们证明P上的每一个同余都可以用这样的同余对进行刻画. 事实上,P上的同余格格同构于P上同余对构成的集合. 最后, 我们刻画P上的群同余.

英文摘要:

      Let $P=E\bowtie G$ be a Zappa-Sz\'{e}p product of a semilattice $E$ with an identity and a group $G$. In this paper, we first introduce the concept of congruence pairs for $P$, and then prove that every congruence on $P$ can be described by such a congruence pair. In fact the congruence lattice on $P$ is lattice-isomorphic to the set of all congruence pairs for $P$. Finally, we characterize group congruences on $P$.