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Ilinka Dimitrova

Publications and source records attributed to Ilinka Dimitrova.

17 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 $\alpha$ is called fence-decreasing if $x\alpha \preceq x$ for all $x$ in the domain of $\alpha$, and fence-preserving if $x\prec y$ implies $x\alpha \preceq y\alpha$ for all $x$ and $y$ in the domain of $\alpha$. 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.

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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 $\mathcal{PORD}(n,r) =\{ \alpha\in \mathcal{PORD}_{n} :\lvert \text{im}(\alpha) \rvert \leq r\}$ and $\mathcal{IORD}(n,r)=\{ \alpha \in \mathcal{IORD}_{n} :\lvert \text{im}(\alpha )\rvert \leq r\}$. 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 $\mathcal{IORD}(n,r)$.

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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 $$\mathcal{ORD}(n,r) =\{\alpha \in \mathcal{ORD}_{n}\, :\, \lvert \textrm{im}(\alpha )\rvert \leq r\}.$$ 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 $3\leq r\leq n-1$.

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arXiv2025

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 $\mathcal{PMD}(n,r) =\{ \alpha\in \mathcal{PMD}_{n} : |\textrm{im}(\alpha)| \leq r\}$ and $\mathcal{IMD}(n,r)=\{ \alpha \in \mathcal{IMD}_{n} :|\textrm{im}(\alpha)| \leq r\}$. 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 $2\leq r\leq n$.

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arXiv2024

Presentations for monoids of partial endomorphisms of a star graph

In this paper, we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to exhibit a presentation for each of them.

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arXiv2024

On partial endomorphisms of a star graph

In this paper we consider the monoids of all partial endomorphisms, of all partial weak endomorphisms, of all injective partial endomorphisms, of all partial strong endomorphisms and of all partial strong weak endomorphisms of a star graph with a finite number of vertices. Our main objective is to determine their ranks. We also describe their Green's relations, calculate their cardinalities and study their regularity.

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arXiv2023

On monoids of endomorphisms of a cycle graph

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities.

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arXiv2023

On three submonoids of the dihedral inverse monoid on a finite set

In this paper we consider three submonoids of the dihedral inverse monoid $\mathcal{DI}_n$, namely its submonoids $\mathcal{OPDI}_n$, $\mathcal{MDI}_n$ and $\mathcal{ODI}_n$ of all orientation-preserving, monotone and order-preserving transformations, respectively. For each of these three monoids, we compute the cardinal, give descriptions of Green's relations and determine the rank.

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arXiv2022

The rank of the inverse semigroup of all partial automorphisms on a finite crown

For $n \in \mathbb{N}$, let $[n] = \{1, 2, \ldots, n\}$ be an $n$ - element set. As usual, we denote by $I_n$ the symmetric inverse semigroup on $[n]$, i.e. the partial one-to-one transformation semigroup on $[n]$ under composition of mappings. The crown (cycle) $\cal{C}_n$ is an $n$-ordered set with the partial order $\prec$ on $[n]$, where the only comparabilities are $$1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~\mbox{ or }~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1.$$ We say that a transformation $\alpha \in I_n$ is order-preserving if $x \prec y$ implies that $x\alpha \prec y\alpha$, for all $x, y $ from the domain of $\alpha$. In this paper, we study the inverse semigroup $IC_n$ of all partial automorphisms on a finite crown $\cal{C}_n$. We consider the elements, determine a generating set of minimal size and calculate the rank of $IC_n$.

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arXiv2021

On the semigroup of all partial fence-preserving injections on a finite set

For $n \in \mathbb{N}$, let $X_n = \{a_1, a_2, \ldots, a_n\}$ be an $n$ - element set and let $\textbf{F} = (X_n; <_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is \textit{fence-preserving} if $x <_f y$ implies that $x\alpha <_f y\alpha$, for all $x, y$ in the domain of $\alpha$. In this paper, we study the semigroup $PFI_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $IF_n = \{\alpha \in PFI_n : \alpha^{-1}\in PFI_n\}$. Clearly, $IF_n$ is an inverse semigroup and contains all regular elements of $PFI_n.$ We characterize the Green's relations for the semigroup $IF_n$. Further, we prove that the semigroup $IF_n$ is generated by its elements with \rank$\geq n-2$. Moreover, for $n \in 2\mathbb{N}$ we find the least generating set and calculate the rank of $IF_n$.

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arXiv2021

On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range

Let $X$ be an infinite linearly ordered set and let $Y$ be a nonempty subset of $X$. We calculate the relative rank of the semigroup $OP(X,Y)$ of all orientation-preserving transformations on $X$ with restricted range $Y$ modulo the semigroup $O(X,Y)$ of all order-preserving transformations on $X$ with restricted range $Y$. For $Y = X$, we characterize the relative generating sets of minimal size.

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arXiv2021

Partial Automorphisms and Injective Partial Endomorphisms of a Finite Undirected Path

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids $IEnd(P_n)$ and $PAut(P_n)$ of all injective partial endomorphisms and of all partial automorphisms of the undirected path $P_n$ with $n$ vertices. We also describe Green's relations of $PAut(P_n)$ and $IEnd(P_n)$ and calculate their cardinals.

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arXiv2020

On relative ranks of finite transformation semigroups with restricted range

In this paper, we determine the relative rank of the semigroup $T(X,Y)$ of all transformations on a finite chain $X$ with restricted range $Y \subseteq X$ modulo the set $OP(X,Y)$ of all orientation-preserving transformation in $T(X,Y)$. Moreover, we state the relative rank of the semigroup $OP(X,Y)$ modulo the set $O(X,Y)$ of all order-preserving transformations in $OP(X,Y)$. In both cases we characterize the minimal relative generating sets.

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arXiv2018

Ranks of monoids of endomorphisms of a finite undirected path

In this paper we study the widely considered endomorphisms and weak endomorphisms of a finite undirected path from monoid generators perspective. Our main aim is to determine the ranks of the monoids $wEnd P_n$ and $End P_n$ of all weak endomorphisms and all endomorphisms of the undirected path $P_n$ with $n$ vertices. We also consider strong and strong weak endomorphisms of $P_n$.

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arXiv2015

A note on generators of the endomorphism semigroup of an infinite countable chain

In this note, we consider the semigroup $O(X)$ of all order endomorphisms of an infinite chain $X$ and the subset $J$ of $O(X)$ of all transformations $\alpha$ such that $|Im(\alpha)|=|X|$. For an infinite countable chain $X$, we give a necessary and sufficient condition on $X$ for $O(X) = \langle J \rangle$ to hold. We also present a sufficient condition on $X$ for $O(X) = \langle J \rangle$ to hold, for an arbitrary infinite chain $X$.

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