Semiring arising as Lattice of Groupsemirings

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,+,· ) where (G,· ) is a group and $(G,+)$ is a left zero semigroup. We consider construction of semirings from classes of group semirings _α :α∈ D \ indexed by a distributive lattice D. It is shown that if S=∪_α \ is a strong distributive lattice of group semirings G_α then the multiplicative semigroup (S,·) of the semiring (S,+,·) is a Clifford semigroup and the additive semigroup $(S,+)$ is a left normal band. Further in this case all the groups G_α are mutually isomorphic.