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Some Properties Of The Semigroup $Pg_Y(X)$: Green's Relations, Ideals, Isomorphism Theorems And Ranks, Worachead Sommanee
Some Properties Of The Semigroup $Pg_Y(X)$: Green's Relations, Ideals, Isomorphism Theorems And Ranks, Worachead Sommanee
Turkish Journal of Mathematics
Let $T(X)$ be the full transformation semigroup on the set $X$. For a fixed nonempty subset $Y$ of $X$, let \begin{equation*} PG_Y(X) = \{\alpha\in T(X) : \alpha _Y\in G(Y)\} \end{equation*} where $G(Y)$ is the permutation group on $Y$. It is known that $PG_Y(X)$ is a regular subsemigroup of $T(X)$. In this paper, we give a simpler description of Green's relations and characterize the ideals of $PG_Y(X)$. Moreover, we prove some isomorphism theorems for $PG_Y(X)$. For finite sets, we investigate the cardinalities of $PG_Y(X)$ and of its subsets of idempotents, and we also calculate their ranks.
Ranks Of Nilpotent Subsemigroups Of Order-Preserving And Decreasing Transformation Semigroups, Emrah Korkmaz, Hayrullah Ayik
Ranks Of Nilpotent Subsemigroups Of Order-Preserving And Decreasing Transformation Semigroups, Emrah Korkmaz, Hayrullah Ayik
Turkish Journal of Mathematics
Let $\mathcal{C}_{n}$ be the semigroup of all order-preserving and decreasing transformations on $X=\{1,\ldots ,n\}$ under its natural order, and let $N(\mathcal{C}_{n})$ be the subsemigroup of all nilpotent elements of $\mathcal{C}_{n}$. For $1\leq r \leq n-1$, let \begin{eqnarray*} N(\mathcal{C}_{n,r})&=&\{ \alpha\in N(\mathcal{C}_{n}) : \lvert im(\alpha)\rvert \leq r\} ,\\ N_{r}(\mathcal{C}_{n})&=&\{\alpha\in N\mathcal({C}_{n}):\alpha\mbox{ is an } m\mbox{-potent for any } 1\leq m\leq r\} . \end{eqnarray*} In this paper we find the cardinality and the rank of the subsemigroup $N(\mathcal{C}_{n,r})$ of $\mathcal{C}_{n}$. Moreover, we show that the set $N_{r}(\mathcal{C}_{n})$ is a subsemigroup of $N(\mathcal{C}_{n})$ and then, we find a lower bound for the rank of $N_{r}(\mathcal{C}_{n})$.
Quasi-Idempotent Ranks Of The Proper Ideals In Finite Symmetric Inverse Semigroups, Leyla Bugay
Quasi-Idempotent Ranks Of The Proper Ideals In Finite Symmetric Inverse Semigroups, Leyla Bugay
Turkish Journal of Mathematics
Let $I_{n}$ and $S_{n}$ be the symmetric inverse semigroup and the symmetric group on a finite chain $X_{n}=\{1,\ldots ,n \}$, respectively. Also, let $I_{n,r}= \{ \alpha \in I_{n}: im(\alpha) \leq r\}$ for $1\leq r\leq n-1$. For any $\alpha\in I_n$, if $\alpha\neq \alpha^2=\alpha^4$ then $\alpha$ is called a quasi-idempotent. In this paper, we show that the quasi-idempotent rank of $I_{n,r}$ (both as a semigroup and as an inverse semigroup) is $\binom{n}{2}$ if $r=2$, and $\binom{n}{r}+1$ if $r\geq 3$. The quasi-idempotent rank of $I_{n,1}$ is $n$ (as a semigroup) and $n-1$ (as an inverse semigroup).