Turkish Journal of Mathematics
DOI
10.3906/mat-2104-22
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.
Keywords
Green's relations, ideal, isomorphism theorem, rank
First Page
1789
Last Page
1800
Recommended Citation
SOMMANEE, WORACHEAD
(2021)
"Some Properties of the Semigroup $PG_Y(X)$: Green's Relations, Ideals, Isomorphism Theorems and Ranks,"
Turkish Journal of Mathematics: Vol. 45:
No.
4, Article 24.
https://doi.org/10.3906/mat-2104-22
Available at:
https://journals.tubitak.gov.tr/math/vol45/iss4/24
