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

Abstract:

      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 $P$.