Semiring arising as Lattice of Groupsemirings

Abstract

Much study has been done on semigroups which are unions of groups. There are several ways in which a union of groups can be made into a semigroup in which each of the component groups arises as subgroups of the constructed semigroup. An important class of such unions is a semilattice of groups. Group semirings are semirings $(G,+,\cdot )$ where $(G,\cdot )$ is a group and $(G,+)$ is a left zero semigroup. We consider construction of semirings from classes of group semirings $\{G_{\alpha }:\alpha \in D\}$ indexed by a distributive lattice $D$. It is shown that if $S=\cup \{G_{\alpha }\}$ is a strong distributive lattice of group semirings $G_{\alpha }$ then the multiplicative semigroup $(S,\cdot )$ of the semiring $(S,+,\cdot )$ is a Clifford semigroup and the additive semigroup $(S,+)$ is a left normal band. Further in this case all the groups $G_{\alpha }$ are mutually isomorphic.