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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2604.22007 |
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| _version_ | 1866914503401144320 |
|---|---|
| author | Andrenšek, Luka |
| author_facet | Andrenšek, Luka |
| contents | We investigate equations in Kiselman's semigroup $K_n$, generated by $a_1, \dots, a_n$. Let $f$ denote the zero element of $K_n$. We prove that if $y \in K_n$ lies in the subsemigroup generated by $a_2, \dots, a_n$, then $x y = f$ implies $x = f$. In contrast, the equation $x a_1 = f$ admits non-trivial solutions. We describe the solution set of this equation, show that its cardinality is $1 + |K_{n-1}|$, and study its algebraic structure. Moreover, we show that $|K_{2n+1}|$ is even, whereas $|K_{2n}|$ is odd. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22007 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Zero Cancellation and Equation Structure in Kiselman's Semigroup Andrenšek, Luka Group Theory We investigate equations in Kiselman's semigroup $K_n$, generated by $a_1, \dots, a_n$. Let $f$ denote the zero element of $K_n$. We prove that if $y \in K_n$ lies in the subsemigroup generated by $a_2, \dots, a_n$, then $x y = f$ implies $x = f$. In contrast, the equation $x a_1 = f$ admits non-trivial solutions. We describe the solution set of this equation, show that its cardinality is $1 + |K_{n-1}|$, and study its algebraic structure. Moreover, we show that $|K_{2n+1}|$ is even, whereas $|K_{2n}|$ is odd. |
| title | Zero Cancellation and Equation Structure in Kiselman's Semigroup |
| topic | Group Theory |
| url | https://arxiv.org/abs/2604.22007 |