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[2304.12893] Semigroup algorithmic problems in metabelian groups
source link: https://arxiv.org/abs/2304.12893
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Mathematics > Group Theory
[Submitted on 25 Apr 2023]
Semigroup algorithmic problems in metabelian groups
We consider semigroup algorithmic problems in finitely generated metabelian groups. Our paper focuses on three decision problems introduced by Choffrut and Karhumäki (2005): the Identity Problem (does a semigroup contain a neutral element?), the Group Problem (is a semigroup a group?) and the Inverse Problem (does a semigroup contain the inverse of a generator?). We show that all three problems are decidable for finitely generated sub-semigroups of finitely generated metabelian groups. In particular, we establish a correspondence between polynomial semirings and sub-semigroups of metabelian groups using an interaction of graph theory, convex polytopes, algebraic geometry and number theory.
Since the Semigroup Membership problem (does a semigroup contain a given element?) is known to be undecidable in finitely generated metabelian groups, our result completes the decidability characterization of semigroup algorithmic problems in metabelian groups.
| Comments: | 52 pages including appendices, 28 figures |
| Subjects: | Group Theory (math.GR); Discrete Mathematics (cs.DM); Algebraic Geometry (math.AG); Rings and Algebras (math.RA) |
| Cite as: | arXiv:2304.12893 [math.GR] |
| (or arXiv:2304.12893v1 [math.GR] for this version) | |
| https://doi.org/10.48550/arXiv.2304.12893 |
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