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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.10103 |
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| _version_ | 1866929244438790144 |
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| author | Rajan, A. R. Sheena, S. Preenu, C. S. |
| author_facet | Rajan, A. R. Sheena, S. Preenu, C. S. |
| contents | 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 $\{G_α:α\in D \}$ indexed by a distributive lattice $D$. It is shown that if $S=\cup\{G_α\}$ is a strong distributive lattice of group semirings $G_α$ 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 $G_α$ are mutually isomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_10103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semiring arising as Lattice of Groupsemirings Rajan, A. R. Sheena, S. Preenu, C. S. Group Theory Rings and Algebras 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 $\{G_α:α\in D \}$ indexed by a distributive lattice $D$. It is shown that if $S=\cup\{G_α\}$ is a strong distributive lattice of group semirings $G_α$ 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 $G_α$ are mutually isomorphic. |
| title | Semiring arising as Lattice of Groupsemirings |
| topic | Group Theory Rings and Algebras |
| url | https://arxiv.org/abs/2402.10103 |