Finite coverings of semigroups and related structures
For a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2023-09-01
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| Series: | International Journal of Group Theory |
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| Online Access: | https://ijgt.ui.ac.ir/article_26750_d18569ae4418178d499f00b8dc03e96e.pdf |
| _version_ | 1811225581164953600 |
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| author | Casey Donoven Luise-Charlotte Kappe |
| author_facet | Casey Donoven Luise-Charlotte Kappe |
| author_sort | Casey Donoven |
| collection | DOAJ |
| description | For a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an inverse semigroup with covering number $n$, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well. |
| first_indexed | 2024-04-12T09:09:45Z |
| format | Article |
| id | doaj.art-c8c80584add84fd9b3245fdce0973479 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-04-12T09:09:45Z |
| publishDate | 2023-09-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-c8c80584add84fd9b3245fdce09734792022-12-22T03:39:01ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692023-09-0112320522210.22108/ijgt.2022.131538.175926750Finite coverings of semigroups and related structuresCasey Donoven0Luise-Charlotte Kappe1Department of Mathematics, Montana State University, Havre, MT, 59501, USADepartment of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USAFor a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an inverse semigroup with covering number $n$, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well.https://ijgt.ui.ac.ir/article_26750_d18569ae4418178d499f00b8dc03e96e.pdfsemigroupcovering numberinverse semigroupmonoid |
| spellingShingle | Casey Donoven Luise-Charlotte Kappe Finite coverings of semigroups and related structures International Journal of Group Theory semigroup covering number inverse semigroup monoid |
| title | Finite coverings of semigroups and related structures |
| title_full | Finite coverings of semigroups and related structures |
| title_fullStr | Finite coverings of semigroups and related structures |
| title_full_unstemmed | Finite coverings of semigroups and related structures |
| title_short | Finite coverings of semigroups and related structures |
| title_sort | finite coverings of semigroups and related structures |
| topic | semigroup covering number inverse semigroup monoid |
| url | https://ijgt.ui.ac.ir/article_26750_d18569ae4418178d499f00b8dc03e96e.pdf |
| work_keys_str_mv | AT caseydonoven finitecoveringsofsemigroupsandrelatedstructures AT luisecharlottekappe finitecoveringsofsemigroupsandrelatedstructures |