Turkish Journal of Mathematics
Abstract
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.
DOI
10.3906/mat-2104-22
Keywords
Green's relations, ideal, isomorphism theorem, rank
First Page
1789
Last Page
1800
Recommended Citation
SOMMANEE, W (2021). Some Properties of the Semigroup $PG_Y(X)$: Green's Relations, Ideals, Isomorphism Theorems and Ranks. Turkish Journal of Mathematics 45 (4): 1789-1800. https://doi.org/10.3906/mat-2104-22
