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...
| Main Authors: | , |
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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 |
| Subjects: | |
| Online Access: | https://ijgt.ui.ac.ir/article_26750_d18569ae4418178d499f00b8dc03e96e.pdf |
| Summary: | 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. |
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| ISSN: | 2251-7650 2251-7669 |