In this paper we characterize the structure of (minimal) generating sets of the subsemigroups $PODI_{n,r}=\{ \alpha \in PODI_{n}:|\im(\alpha)|\leq r\}$, $POI_{n,r}=\{ \alpha \in POI_{n}: |\im(\alpha)|\leq r\}$, and $E_{n,r}=\{ \id_{A}\in I_{n}:A\subseteq X_n\mbox{ and }|A|\leq r\}$ where $id_{A}$ is the identity map on $A\subseteq X_n$ for $0\leq r\leq n-1$.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $I_{n}$ be the symmetric inverse semigroup, and let $PODI_{n}$ and $POI_{n}$ be its subsemigroups of monotone partial bijections and of isotone partial bijections on $X_{n}=\{1,\ldots ,n\}$ under its natural order, respectively.
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Key Findings & Data:Let $I_{n}$ be the symmetric inverse semigroup, and let $PODI_{n}$ and $POI_{n}$ be its subsemigroups of monotone partial bijections and of isotone partial bijections on $X_{n}=\{1,\ldots ,n\}$ under its natural order, respectively.
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Conclusion & Impact:In this paper we characterize the structure of (minimal) generating sets of the subsemigroups $PODI_{n,r}=\{ \alpha \in PODI_{n}:|\im(\alpha)|\leq r\}$, $POI_{n,r}=\{ \alpha \in POI_{n}: |\im(\alpha)|\leq r\}$, and $E_{n,r}=\{ \id_{A}\in I_{n}:A\subseteq X_n\mbox{ and }|A|\leq r\}$ where $id_{A}$ is the identity map on $A\subseteq X_n$ for $0\leq r\leq n-1$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Partial bijection · isotone/antitone/monotone map · (minimal) generating set
Abstract
Let $I_{n}$ be the symmetric inverse semigroup, and let $PODI_{n}$ and $POI_{n}$ be its subsemigroups of monotone partial bijections and of isotone partial bijections on $X_{n}=\{1,\ldots ,n\}$ under its natural order, respectively. In this paper we characterize the structure of (minimal) generating sets of the subsemigroups $PODI_{n,r}=\{ \alpha \in PODI_{n}:|\im(\alpha)|\leq r\}$, $POI_{n,r}=\{ \alpha \in POI_{n}: |\im(\alpha)|\leq r\}$, and $E_{n,r}=\{ \id_{A}\in I_{n}:A\subseteq X_n\mbox{ and }|A|\leq r\}$ where $id_{A}$ is the identity map on $A\subseteq X_n$ for $0\leq r\leq n-1$.