Presentations and word problem for strong semilattices of semigroups
Let \(I\) be a semilattice, and \(S_i\) \((i\in I)\) be a family of disjoint semigroups. Then we prove that the strong semilattice \(S=\mathcal{S} [I,S_i,\phi _{j,i}]\) of semigroups \(S_i\) with homomorphisms \(\phi _{j,i}:S_j\rightarrow S_i\) (\(j\geq i\)) is finitely presented if and only if \(I\...
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| Date: | 2018 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/943 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | Let \(I\) be a semilattice, and \(S_i\) \((i\in I)\) be a family of disjoint semigroups. Then we prove that the strong semilattice \(S=\mathcal{S} [I,S_i,\phi _{j,i}]\) of semigroups \(S_i\) with homomorphisms \(\phi _{j,i}:S_j\rightarrow S_i\) (\(j\geq i\)) is finitely presented if and only if \(I\) is finite and each \(S_i\) \((i\in I)\) is finitely presented. Moreover, for a finite semilattice \(I\), \(S\) has a soluble word problem if and only if each \(S_i\) \((i\in I)\) has a soluble word problem. Finally, we give an example of non-automatic semigroup which has a soluble word problem. |
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