Free idempotent generated semigroups: The word problem and structure via gain graphs
| Autor: | Igor Dolinka |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Intersection (set theory) Semigroup General Mathematics Structure (category theory) Group Theory (math.GR) Decidability Word problem (mathematics education) Set (abstract data type) Primary 20M05 Secondary 20F10 Idempotence FOS: Mathematics Coset Mathematics - Group Theory Mathematics |
| Zdroj: | Israel Journal of Mathematics. 245:347-387 |
| ISSN: | 1565-8511 0021-2172 |
| DOI: | 10.1007/s11856-021-2214-1 |
| Popis: | Building on the previous extensive study of Yang, Gould and the present author, we provide a more precise insight into the group-theoretical ramifications of the word problem for free idempotent generated semigroups over finite biordered sets. We prove that such word problems are in fact equivalent to the problem of computing intersections of cosets of certain subgroups of direct products of maximal subgroups of the free idempotent generated semigroup in question, thus providing decidability of those word problems under group-theoretical assumptions related to the Howson property and the coset intersection property. We also provide a basic sketch of the global semigroup-theoretical structure of an arbitrary free idempotent generated semigroup, including the characterisation of Green's relations and the key parameters of non-regular $\mathscr{D}$-classes. In particular, we prove that all Sch\"utzenberger groups of $\mathsf{IG}(\mathcal{E})$ for a finite biordered set $\mathcal{E}$ must be among the divisors of the maximal subgroups of $\mathsf{IG}(\mathcal{E})$. Comment: Israel Journal of Mathematics (to appear). 28 pages; some minor text overlap with arXiv:1802.02420 and arXiv:1412.5167 in the preliminary section (because of the closely related topic) |
| Databáze: | OpenAIRE |
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