Green’s relations for 2 × 2 matrices over linearly ordered abelian groups
We consider semigroups of 2 Ã 2 matrices over linearly ordered abelian groups with respect to multiplication, which is defined similarly to tropical algebra. We study Greenâs relations on such semigroups. In particular, we describe the R-, L- and H-classes of such semigroups and give a simple criter...
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Estonian Academy Publishers
2025-02-01
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| Series: | Proceedings of the Estonian Academy of Sciences |
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| Online Access: | https://kirj.ee/wp-content/plugins/kirj/pub/proc-1-2025-62-70_20250207125248.pdf |
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| author | Marilyn Kutti Valdis Laan |
| author_facet | Marilyn Kutti Valdis Laan |
| author_sort | Marilyn Kutti |
| collection | DOAJ |
| description | We consider semigroups of 2 Ã 2 matrices over linearly ordered abelian groups with respect to multiplication, which is defined similarly to tropical algebra. We study Greenâs relations on such semigroups. In particular, we describe the R-, L- and H-classes of such semigroups and give a simple criterion for determining whether two matrices are D-related. We prove that the D-relation coincides with the J-relation. We also study maximal subgroups of such semigroups. It turns out that if the abelian group is divisible, then these maximal subgroups can have two different forms. |
| format | Article |
| id | doaj-art-16b39ad5b8ae49888980d4867be03f50 |
| institution | Kabale University |
| issn | 1736-6046 1736-7530 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | Estonian Academy Publishers |
| record_format | Article |
| series | Proceedings of the Estonian Academy of Sciences |
| spelling | doaj-art-16b39ad5b8ae49888980d4867be03f502025-02-10T10:10:04ZengEstonian Academy PublishersProceedings of the Estonian Academy of Sciences1736-60461736-75302025-02-017416270https://doi.org/10.3176/proc.2025.1.06https://doi.org/10.3176/proc.2025.1.06Green’s relations for 2 × 2 matrices over linearly ordered abelian groupsMarilyn Kutti0Valdis Laan1Institute of Mathematics and Statistics, University of Tartu, Narva mnt. 18, 51009 Tartu, EstoniaInstitute of Mathematics and Statistics, University of Tartu, Narva mnt. 18, 51009 Tartu, EstoniaWe consider semigroups of 2 Ã 2 matrices over linearly ordered abelian groups with respect to multiplication, which is defined similarly to tropical algebra. We study Greenâs relations on such semigroups. In particular, we describe the R-, L- and H-classes of such semigroups and give a simple criterion for determining whether two matrices are D-related. We prove that the D-relation coincides with the J-relation. We also study maximal subgroups of such semigroups. It turns out that if the abelian group is divisible, then these maximal subgroups can have two different forms.https://kirj.ee/wp-content/plugins/kirj/pub/proc-1-2025-62-70_20250207125248.pdfalgebramatrix semigrouplinearly ordered abelian groupgreen’s relationsmaximal subgroupcompletely simple semigroup |
| spellingShingle | Marilyn Kutti Valdis Laan Green’s relations for 2 × 2 matrices over linearly ordered abelian groups Proceedings of the Estonian Academy of Sciences algebra matrix semigroup linearly ordered abelian group green’s relations maximal subgroup completely simple semigroup |
| title | Green’s relations for 2 × 2 matrices over linearly ordered abelian groups |
| title_full | Green’s relations for 2 × 2 matrices over linearly ordered abelian groups |
| title_fullStr | Green’s relations for 2 × 2 matrices over linearly ordered abelian groups |
| title_full_unstemmed | Green’s relations for 2 × 2 matrices over linearly ordered abelian groups |
| title_short | Green’s relations for 2 × 2 matrices over linearly ordered abelian groups |
| title_sort | green s relations for 2 2 matrices over linearly ordered abelian groups |
| topic | algebra matrix semigroup linearly ordered abelian group green’s relations maximal subgroup completely simple semigroup |
| url | https://kirj.ee/wp-content/plugins/kirj/pub/proc-1-2025-62-70_20250207125248.pdf |
| work_keys_str_mv | AT marilynkutti greensrelationsfor22matricesoverlinearlyorderedabeliangroups AT valdislaan greensrelationsfor22matricesoverlinearlyorderedabeliangroups |