arXiv subjects

Hayrullah Ayık

Publications and source records attributed to Hayrullah Ayık.

7 recordsLinked to original sources

arXiv2026

On the semigroups of fence-decreasing and fence-preserving transformations on a finite fence

For a natural number $n$, a fence $[n]=\{1\prec 2\succ 3\prec 4\succ 5\prec \cdots n\}$ is a partial ordered set. A partial transformation $α$ is called fence-decreasing if $xα\preceq x$ for all $x$ in the domain of $α$, and fence-preserving if $x\prec y$ implies $xα\preceq yα$ for all $x$ and $y$ in the domain of $α$. In this paper, we consider the monoids $\mathcal{DF}_{n}$ ($\mathcal{PDF}_{n})$ of all fence-decreasing full (partial) transformations as well as the monoid $\mathcal{PCF}_{n}$ of all fence-preserving transformations of $\mathcal{PDF}_{n}$. For these three monoids and some of their ideals, we determine the unique minimal generating set. Moreover, we calculate the rank of $\mathcal{DF}_{n}$, $\mathcal{PDF}_{n}$, and $\mathcal{PCF}_{n}$. Additional, we provide several combinatorial results concerning these three monoids.

math.RA↗

arXiv2026

On certain semigroups of finite monotone and order-decreasing partial transformations

Let $\mathcal{PMD}_{n}$ be the semigroup consisting of all monotone and order-decreasing partial transformations, and let $\mathcal{IMD}_{n}$ be the subsemigroup of $\mathcal{PMD}_{n}$ consisting of all injective monotone and order-decreasing transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. For $2\leq r\leq n$, let $\mathcal{PMD}(n,r) =\{ α\in \mathcal{PMD}_{n} : |\textrm{im}(α)| \leq r\}$ and $\mathcal{IMD}(n,r)=\{ α\in \mathcal{IMD}_{n} :|\textrm{im}(α)| \leq r\}$. In this paper, we determine the cardinalities, maximal subsemigroups and ranks of $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$, and moreover, we verify that the semigroups $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$ are non-regular but abundant for any $2\leq r\leq n$.

math.RA↗

arXiv2025

Combinatorial results for zero-divisors regarding right zero elements of order-preserving transformations

For any positive integer $n$, let $\mathcal{O}_{n}$ be the semigroup of all order-preserving full transformations on $X_{n}=\{1<\cdots <n\}$. For any $1\leq k\leq n$, let $π_{k}\in \mathcal{O}_{n}$ be the constant map defined by $xπ_{k}=k$ for all $x\in X_{n}$. In this paper, we introduce and study the sets of left, right, and two-sided zero-divisors of $π_{k}$: \begin{eqnarray*} \mathsf{L}_{k} &=& \{ α\in \mathcal{O}_{n}:αβ=π_{k} \mbox{ for some }β\in \mathcal{O}_{n} \setminus\{π_{k}\} \}, \mathsf{R}_{k} &=& \{ α\in \mathcal{O}_{n}:γα=π_{k} \mbox{ for some }\ γ\in \mathcal{O}_{n}\setminus\{π_{k}\} \}, \ \mbox{and} \ \mathsf{Z}_{k}=\mathsf{L}_{k}\cap \mathsf{R}_{k}. \end{eqnarray*} We determine the structures and cardinalities of $\mathsf{L}_{k}$, $\mathsf{R}_{k}$ and $\mathsf{Z}_{k}$ for each $1\leq k\leq n$. Furthermore, we compute the ranks of $\mathsf{R}_{1}$,\, $\mathsf{R}_{n}$,\, $\mathsf{Z}_{1}$,\, $\mathsf{Z}_{n}$ and $\mathsf{L}_{k}$ for each $1\leq k\leq n$, because these are significant subsemigroups of $\mathcal{O}_{n}$.

math.RA↗

arXiv2025

On certain semigroups of finite oriented and order-decreasing partial transformations

Let $\mathcal{PORD}_{n}$ be the semigroup consisting of all oriented and order-decreasing partial transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. Let $\mathcal{IORD}_{n}$ be the subsemigroup of $\mathcal{PORD}_{n}$ consisting of all injective transformations of $\mathcal{PORD}_{n}$. For $2\leq r\leq n$, let $\mathcal{PORD}(n,r) =\{ α\in \mathcal{PORD}_{n} :\lvert \text{im}(α) \rvert \leq r\}$ and $\mathcal{IORD}(n,r)=\{ α\in \mathcal{IORD}_{n} :\lvert \text{im}(α)\rvert \leq r\}$. In this paper, we determine some minimal generating sets and ranks of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$, and moreover, we characterize the maximal subsemigroups of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$.

math.RA↗

arXiv2025

On certain subsemigroups of finite oriented and order-decreasing full transformations

Let $\mathcal{ORD}_{n}$ be the semigroup consisting of all oriented and order-decreasing full transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$, and for $1\leq r\leq n-1$, let $$\mathcal{ORD}(n,r) =\{α\in \mathcal{ORD}_{n}\, :\, \lvert \textrm{im}(α)\rvert \leq r\}.$$ In this paper, we determine the cardinality of $\mathcal{ORD}(n,r)$ and the number of nilpotent elements of $\mathcal{ORD}(n,r)$, we find a minimal generating set and the rank of $\mathcal{ORD}(n,r)$, and moreover, we characterize all maximal subsemigroups of $\mathcal{ORD}(n,r)$ for each $3\leq r\leq n-1$.

math.RA↗

arXiv2025

On monoids of monotone partial transformations of a finite chain whose domains and ranges are intervals

In this note, we consider the monoid $\mathcal{PIM}_{n}$ of all partial monotone transformations on a chain with $n$ elements whose domains and ranges are intervals and its submonoid $\mathcal{IM}_{n}$ constituted by the full transformations. For both of these monoids, our aim is to determine their cardinalities and ranks and define them by means of presentations. We also calculate the number of nilpotent elements of $\mathcal{PIM}_{n}$.

math.RA↗

arXiv2025

On the monoid of partial order-preserving transformations of a finite chain whose domains and ranges are intervals

In this paper, we consider the monoid $\mathcal{PIO}_{n}$, of all partial order-preserving transformations on a chain with $n$ elements whose domains and ranges are intervals, along with its submonoid $\mathcal{PIO}_{n}^-$ of order-decreasing transformations. Our main aim is to give presentations for $\mathcal{PIO}_{n}^-$ and $\mathcal{PIO}_{n}$. Moreover, for both monoids, we describe regular elements and determine their ranks, cardinalities and the numbers of idempotents and nilpotents.

math.RA↗