arXiv subjects

Muhammad Mansur Zubairu

Publications and source records attributed to Muhammad Mansur Zubairu.

4 recordsLinked to original sources

arXiv2025

The monoid of monotone and decreasing partial transformations on a finite chain

In this article, we consider the monoid of all monotone and order-decreasing partial transformations denoted as $\mathcal{DORP}_{n}$ on an $n$ ordered chain $[n]=\{1, \ldots,n\}$, its two-sided ideal $I(n,p)= \{ρ\in \mathcal{DORP}_{n} : \, |Im \, ρ| \leq p\}$ and the Rees quotient ${RQ}_{p}(n)$ of the ideal $I(n,p)$. We compute the order of the monoid $\mathcal{DORP}_{n}$ and show that for any semigroup $S$ in $\{\mathcal{DORP}_{n}, \, I(n,p), \, {RQ}_{p}(n)\}$, $S$ is abundant for all values of $n$. In particular, we show that the Rees quotient ${RQ}_{p}(n)$, is a non-regular $0-*$bisimple abundant semigroup. In addition, we compute the ranks of the Rees quotient ${RQ}_{p}(n)$ and the two-sided ideal $I(n,p)$. Finally, the rank of $\mathcal{DORP}_{n}$ is determined to be $3n-2$.

math.GR↗

arXiv2025

On the small Schröder semigroup $\mathcal{SS}^{\prime}_{n}$

Let $[n]$ be a finite $n-$chain $\{1, 2, \dots, n\}$, and let $\mathcal{LS}_{n}$ be the Schröder monoid, consisting of all isotone and order-decreasing partial transformations on $[n]$. Furthermore, let $\mathcal{SS}^{\prime}_{n} = \{α\in \mathcal{LS}_{n} : \, 1\not\in \text{ Dom } α\}$ be the subsemigroup of $\mathcal{LS}_{n}$, consisting of all transformations in $\mathcal{LS}_{n}$, each of whose domain does not contain $1$. For $1 \leq p \leq n$, let $K(n,p) = \{α\in \mathcal{SS}^{\prime}_{n} : \, |Im \, α| \leq p\}$ be the two-sided ideal of $\mathcal{SS}^{\prime}_{n}$. Moreover, let ${RSS}^{\prime}_{n}(p)$ denote the Rees quotient of $K(n,p)$. It is shown in this article that for any $S$ in $\{\mathcal{SS}^{\prime}_{n}, K(n,p), {RSS}^{\prime}_{n}(p)\}$, $S$ is right abundant for all values of $n$, but not left abundant for all $n \geq 2$. In addition, the rank of the Rees quotient ${RSS}^{\prime}_{n}(p)$ is shown to be equal to the rank of the two-sided ideal $K(n,p)$, which is equal to $\binom{n-1}{p-1}+\sum\limits_{k=p}^{n-1}\binom{n-1}{k} \binom{k-1}{p-1}$. Finally, the rank of $\mathcal{SS}^{\prime}_{n}$ is determined to be $3n-4$.

math.GR↗

arXiv2024

On the algebraic structure of the Schröder monoid

Let $[n]$ be a finite chain $\{1, 2, \ldots, n\}$, and let $\mathcal{LS}_{n}$ be the semigroup consisting of all isotone and order-decreasing partial transformations on $[n]$. Moreover, let $\mathcal{SS}_{n} = \{α\in \mathcal{LS}_{n} : \, 1 \in \text{Dom } α\}$ be the subsemigroup of $\mathcal{LS}_{n}$, consisting of all transformations in $\mathcal{LS}_{n}$ each of whose domain contains $1$. For $1 \leq p \leq n$, let $K(n,p) = \{α\in \mathcal{LS}_{n} : \, |\text{Im } \, α| \leq p\}$ and $M(n,p) = \{α\in \mathcal{SS}_{n} : \, |\text{Im } α| \leq p\}$ be the two-sided ideals of $\mathcal{LS}_{n}$ and $\mathcal{SS}_{n}$, respectively. Furthermore, let ${RLS}_{n}(p)$ and ${RSS}_{n}(p)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any $S \in \{\mathcal{SS}_{n}, \mathcal{LS}_{n}, {RLS}_{n}(p), {RSS}_{n}(p)\}$, $S$ is abundant and idempotent generated for all values of $n$. Moreover, the ranks of the Rees quotients ${RLS}_{n}(p)$ and ${RSS}_{n}(p)$ are shown to be equal to the ranks of the two-sided ideals $K(n,p)$ and $M(n,p)$, respectively. Finally, these ranks are computed to be $\sum\limits_{k=p}^{n} \binom{n}{k} \binom{k-1}{p-1}$ and $\binom{n-1}{p-1}2^{n-p}$, respectively.

math.GR↗

arXiv2022

On the combinatorial and rank properties of certain subsemigroups of full contractions of a finite chain

Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{CT}_{n}$ be the semigroup of full contractions on $[n]$. Denote $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$ to be the subsemigroup of order preserving or reversing and the subsemigroup of order preserving full contractions, respectively. It was shown in [17] that the collection of all regular elements (denoted by, Reg$(\mathcal{ORCT}_{n})$ and Reg$(\mathcal{OCT}_{n}$), respectively) and the collection of all idempotent elements (denoted by E$(\mathcal{ORCT}_{n})$ and E$(\mathcal{OCT}_{n}$), respectively) of the subsemigroups $\mathcal{ORCT}_{n}$ and $\mathcal{OCT}_{n}$, respectively are subsemigroups. In this paper, we study some combinatorial and rank properties of these subsemigroups.

math.GR↗