Related papers: Variants of finite full transformation semigroups

Let $T_X$ be the full transformation monoid over a finite set $X$, and fix some $a\in T_X$ of rank $r$. The variant $T_X^a$ has underlying set $T_X$, and operation $f\star g=fag$. We study the congruences of the subsemigroup $P=Reg(T_X^a)$…

Let $a$ be an element of a semigroup $S$. The local subsemigroup of $S$ with respect to $a$ is the subsemigroup $aSa$ of $S$. The variant of $S$ with respect to $a$ is the semigroup with underlying set $S$ and operation $\star_a$ defined by…

Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by $$ T_{E^*}(X)=\{\alpha\in T(X):\text{for all}\ x,y\in X, (x,y)\in…

Let $a$ be a non-invertible transformation of a finite set and let $G$ be a group of permutations on that same set. Then $\genset{G, a}\setminus G$ is a subsemigroup, consisting of all non-invertible transformations, in the 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…

For an element a in a semigroup S the local subsemigroup of S with respect to a is the subsemigroup aSa of S and the variant of S with respect to a is a semigroup with underlying set S with a sandwich operation xy = xay for all x, y in S.…

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…

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…

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…

Suppose that $X$ be a nonempty set. Denote by $\mathcal{T}(X)$ the full transformation semigroup on $X$. For $\varnothing \neq Z\subseteq Y\subseteq X$, let $\mathcal{T}(X,Y,Z)=\{\alpha \in \mathcal{T}(X): Y\alpha \subseteq Z \}$. Then…

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 $T(X,Y)$.…

We construct a family of representations of an arbitrary variant $S_a$ of a semigroup $S$, induced by a given representation of $S$, and investigate properties of such representations and their kernels.

For a positive integer $n$, the full transformation semigroup $T_n$ consists of all self maps of the set $\{1,\ldots,n\}$ under composition. Any finite semigroup $S$ embeds in some $T_n$, and the least such $n$ is called the (minimum…

Fix sets $X$ and $Y$, and write $\mathcal{PT}_{XY}$ for the set of all partial functions $X\to Y$. Fix a partial function $a:Y\to X$, and define the operation $\star_a$ on $\mathcal{PT}_{XY}$ by $f\star_ag=fag$ for…

Let $T(X)$ be the full transformation semigroup on a set $X$ under the composition of functions. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by $$T_{E^*}(X)=\{\alpha\in T(X):\text{for all}\ x,y\in…

We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X=N of natural numbers containing a given subsemigroup W of T(X) where each element of a given set $U$ is a generator of T(X) modulo W. This note…

Let $\Omega$ be a finite set and $T(\Omega)$ be the full transformation monoid on $\Omega$. The rank of a transformation $t\in T(\Omega)$ is the natural number $|\Omega t|$. Given $A\subseteq T(\Omega)$, denote by $\langle A\rangle$ the…

Cross-connection theory propounded by K. S. S. Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant $\mathscr{T}_X^\theta$ of the full transformation semigroup…

Fix (not necessarily distinct) objects $i$ and $j$ of a locally small category $S$, and write $S_{ij}$ for the set of all morphisms $i\to j$. Fix a morphism $a\in S_{ji}$, and define an operation $\star_a$ on $S_{ij}$ by $x\star_ay=xay$ for…

Let $V$ be a vector space and $U$ a fixed subspace of $V$. We denote the semigroup of all linear transformations on $V$ under composition of functions by $L(V)$. In this paper, we study the semigroup of all linear transformations on $V$…