arXiv subjects

Gonca Ayık

Publications and source records attributed to Gonca Ayık.

4 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

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↗