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.