arXiv subjects

Mosarof Sarkar

Publications and source records attributed to Mosarof Sarkar.

8 recordsLinked to original sources

arXiv2023

Semigroups of transformations whose characters belong to a given semigroup

Let $X$ be a nonempty set and $\mathcal{P}=\{X_i\colon i\in I\}$ a partition of $X$. Denote by $T(X)$ the full transformation semigroup on $X$, and $T(X, \mathcal{P})$ the subsemigroup of $T(X)$ consisting of all transformations that preserve $\mathcal{P}$. For every subsemigroup $\mathbb{S}(I)$ of $T(I)$, let $T_{\mathbb{S}(I)}(X,\mathcal{P})$ be the semigroup of all transformations $f\in T(X, \mathcal{P})$ such that $\chi^{(f)}\in \mathbb{S}(I)$, where $\chi^{(f)}\in T(I)$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$. We describe regular and idempotent elements in $T_{\mathbb{S}(I)}(X,\mathcal{P})$, and determine when $T_{\mathbb{S}(I)}(X,\mathcal{P})$ is a regular semigroup [inverse semigroup]. With the assumption that $\mathbb{S}(I)$ contains the identity, we characterize Green's relations on $T_{\mathbb{S}(I)}(X,\mathcal{P})$, describe unit-regular elements in $T_{\mathbb{S}(I)}(X,\mathcal{P})$, and determine when $T_{\mathbb{S}(I)}(X,\mathcal{P})$ is a unit-regular semigroup. We apply these general results to obtain more concrete results for $T(X,\mathcal{P})$.

math.RA↗

arXiv2023

Semigroups of (linear) transformations whose restrictions belong to a given semigroup

Let $T(X)$ (resp. L(V)) be the semigroup of all transformations (resp. linear transformations) of a set $X$ (resp. vector space $V$). For a subset $Y$ of $X$ and a subsemigroup $\mathbb{S}(Y)$ of $T(Y)$, consider the subsemigroup $T_{\mathbb{S}(Y)}(X) = \{f\in T(X)\colon f_{\upharpoonright_Y} \in \mathbb{S}(Y)\}$ of $T(X)$, where $f_{\upharpoonright_Y}\in T(Y)$ agrees with $f$ on $Y$. We give a new characterization for $T_{\mathbb{S}(Y)}(X)$ to be a regular semigroup [inverse semigroup]. For a subspace $W$ of $V$ and a subsemigroup $\mathbb{S}(W)$ of $L(W)$, we define an analogous subsemigroup $L_{\mathbb{S}(W)}(V) = \{f\in L(V) \colon f_{\upharpoonright_W} \in \mathbb{S}(W)\}$ of $L(V)$. We describe regular elements in $L_{\mathbb{S}(W)}(V)$ and determine when $L_{\mathbb{S}(W)}(V)$ is a regular semigroup [inverse semigroup, completely regular semigroup]. If $\mathbb{S}(Y)$ (resp. $\mathbb{S}(W)$) contains the identity of $T(Y)$ (resp. $L(W)$), we describe unit-regular elements in $T_{\mathbb{S}(Y)}(X)$ (resp. $L_{\mathbb{S}(W)}(V)$) and determine when $T_{\mathbb{S}(Y)}(X)$ (resp. $L_{\mathbb{S}(W)}(V)$) is a unit-regular semigroup.

math.GR↗

arXiv2022

Green's relations and unit-regularity for semigroup of transformations whose characters are bijective

Let $X$ be a nonempty set and $\mathcal{P}=\{X_i\colon i\in I\}$ be a partition of $X$. Denote by $T(X, \mathcal{P})$ the semigroup of all transformations of $X$ that preserve $\mathcal{P}$. In this paper, we study the semigroup $\mathcal{B}(X,\mathcal{P})$ of all transformations $f\in T(X, \mathcal{P})$ such that $χ^{(f)}\in {\rm Sym}(I)$, where ${\rm Sym}(I)$ is the symmetric group on $I$ and $χ^{(f)}\colon I \to I$ is the character (map) of $f$ defined by $iχ^{(f)}=j$ whenever $X_if\subseteq X_j$. We describe unit-regular elements in $\mathcal{B}(X,\mathcal{P})$, and determine when $\mathcal{B}(X,\mathcal{P})$ is a unit-regular semigroup. We alternatively prove that $\mathcal{B}(X,\mathcal{P})$ is a regular semigroup. We describe Green's relations on $\mathcal{B}(X,\mathcal{P})$, and prove that $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$ when $\mathcal{P}$ is finite. We also give a necessary and sufficient condition for $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$. We end the paper with a conjecture.

math.GR↗

arXiv2021

On certain semigroups of transformations with an invariant set

Let $X$ be a nonempty set and let $T(X)$ be the full transformation semigroup on $X$. The main objective of this paper is to study the subsemigroup $\overlineΩ(X, Y)$ of $T(X)$ defined by \[\overlineΩ(X, Y) = \{f\in T(X)\colon Yf = Y\},\] where $Y$ is a fixed nonempty subset of $X$. We describe regular elements in $\overlineΩ(X, Y)$ and show that $\overlineΩ(X, Y)$ is regular if and only if $Y$ is finite. We characterize unit-regular elements in $\overlineΩ(X, Y)$ and prove that $\overlineΩ(X, Y)$ is unit-regular if and only if $X$ is finite. We characterize Green's relations on $\overlineΩ(X, Y)$ and prove that $\mathcal{D} =\mathcal{J}$ on $\overlineΩ(X, Y)$ if and only if $Y$ is finite. We also determine ideals of $\overlineΩ(X, Y)$ and investigate its kernel. This paper extends several results appeared in the literature.

math.GR↗

arXiv2021

Unit-regular and semi-balanced elements in various semigroups of transformations

Let $T(X)$ be the full transformation semigroup on a set $X$, and let $L(V)$ be the semigroup under composition of all linear transformations on a vector space $V$ over a field. For a subset $Y$ of $X$ and a subspace $W$ of $V$, consider the semigroups $\overline{T}(X, Y) = \{f\in T(X)\colon Yf \subseteq Y\}$ and $\overline{L}(V, W) = \{f\in L(V)\colon Wf \subseteq W\}$ under composition. We describe unit-regular elements in $\overline{T}(X, Y)$ and $\overline{L}(V, W)$. Using these, we determine when $\overline{T}(X, Y)$ and $\overline{L}(V, W)$ are unit-regular. We prove that $f\in L(V)$ is unit-regular if and only if ${\rm nullity}(f) = {\rm corank}(f)$. We alternatively prove that $L(V)$ is unit-regular if and only if $V$ is finite-dimensional. A semi-balanced semigroup is a transformation semigroup whose all elements are semi-balanced. We give necessary and sufficient conditions for $\overline{T}(X, Y)$, $\overline{L}(V, W)$ and $L(V)$ to be semi-balanced.

math.GR↗

arXiv2021

On unit-regular elements in various monoids of transformations

Let $X$ be an arbitrary set and let $T(X)$ denote the full transformation monoid on $X$. We prove that an element of $T(X)$ is unit-regular if and only if it is semi-balanced. For infinite $X$, we discuss regularity of the submonoid of $T(X)$ consisting of all injective (resp. surjective) transformations. For a partition $\mathcal{P}$ of $X$, we characterize unit-regular elements in the monoid $T(X, \mathcal{P})$, under composition, defined as \[T(X, \mathcal{P}) = \{f\in T(X)\mid (\forall X_i \in \mathcal{P}) (\exists X_j \in \mathcal{P})\; X_i f \subseteq X_j\}.\] We also characterize (unit-)regular elements in various known submonoids of $T(X, \mathcal{P})$.

math.GR↗

arXiv2020

Regular, Unit-regular, and Idempotent elements of semigroups of transformations that preserve a partition

Let $X$ be a set and $\mathcal{T}_X$ be the full transformation semigroup on $X$. For a partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in \mathcal{P}) (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $Σ(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})|(\forall X_i \in \mathcal{P})\; Xf \cap X_i \neq \emptyset\}$, and $Γ(X, \mathcal{P}) = \{f\in \mathcal{T}_X|(\forall X_i\in \mathcal{P})(\exists X_j\in \mathcal{P})\; X_i f = X_j\}$. We characterize unit-regular elements of both $T(X, \mathcal{P})$ and $Σ(X, \mathcal{P})$ for finite $X$. We discuss set inclusion between $Γ(X, \mathcal{P})$ and certain semigroups of transformations preserving $\mathcal{P}$. We characterize and count regular elements and idempotents of $Γ(X, \mathcal{P})$. For finite $X$, we prove that every regular element of $Γ(X, \mathcal{P})$ is unit-regular and also calculate the size of $Γ(X, \mathcal{P})$.

math.GR↗

arXiv2020

On certain Semigroups of Transformations that preserve a partition

Let $X$ be a nonempty set, and let $\mathcal{T}_X$ be the full transformation semigroup on $X$. For a partition $\mathcal{P} = \{X_i \;|\; i\in I\}$ of $X$, we consider the semigroup $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X\;|\; \forall X_i\;\exists X_j,\; X_i f \subseteq X_j\}$, the subsemigroup $Σ(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})\;|\; Xf \cap X_i \neq \emptyset\; \forall X_i\}$, and the group of units $S(X, \mathcal{P})$ of $T(X, \mathcal{P})$. In this paper, we first characterize the elements of $Σ(X, \mathcal{P})$. For a permutation $f$ of finite $X$, we next observe whether there exists a nontrivial partition $\mathcal{P}$ of $X$ such that $f\in S(X, \mathcal{P})$. We then characterize and enumerate the idempotents in the semigroup $Σ(X, \mathcal{P})$ for arbitrary and finite $X$, respectively. We also characterize the elements of $S(X, \mathcal{P})$. For finite $X$, we finally calculate the cardinality of $T(X, \mathcal{P})$, $Σ(X, \mathcal{P})$, and $S(X, \mathcal{P})$.

math.GR↗