Affiliation:
1. School of Mathematics, Yunnan Normal University , Kunming , Yunnan, 650500 , P. R. China
Abstract
Abstract
For a non-empty set
X
X
, denote the full transformation semigroup on
X
X
by
T
(
X
)
T\left(X)
and suppose that
E
E
is an equivalence relation on
X
X
. Evidently,
T
E
∗
(
X
)
=
{
α
∈
T
(
X
)
∣
(
x
,
y
)
∈
E
if and only if
(
x
α
,
y
α
)
∈
E
for all
x
,
y
∈
X
}
{T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X)| \left(x,y)\in E\hspace{0.33em}\hspace{0.1em}\text{if and only if}\hspace{0.1em}\hspace{0.33em}\left(x\alpha ,y\alpha )\in E\hspace{0.33em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}x,y\in X\right\}
is a subsemigroup of
T
(
X
)
T\left(X)
. In this article, we investigate Green relations, Green
∗
\ast
-relations and Green
∼
\sim
-relations, various kinds of regularities,
ℱ
{\mathcal{ {\mathcal F} }}
-abundant and
G
{\mathcal{G}}
-abundant elements and left and right magnifying elements in
T
E
∗
(
X
)
{T}_{{E}^{\ast }}\left(X)
. More specifically, we first obtain the necessary and sufficient conditions under which
ℒ
{\mathcal{ {\mathcal L} }}
(respectively,
ℒ
∗
{{\mathcal{ {\mathcal L} }}}^{\ast }
,
ℒ
˜
\widetilde{{\mathcal{ {\mathcal L} }}}
,
ℛ
{\mathcal{ {\mathcal R} }}
,
ℛ
∗
{{\mathcal{ {\mathcal R} }}}^{\ast }
, and
ℛ
˜
\widetilde{{\mathcal{ {\mathcal R} }}}
) is (left, right) compatible,
ℛ
=
ℛ
∗
{\mathcal{ {\mathcal R} }}={{\mathcal{ {\mathcal R} }}}^{\ast }
or
ℒ
=
ℒ
˜
{\mathcal{ {\mathcal L} }}=\widetilde{{\mathcal{ {\mathcal L} }}}
. Then, we give the sufficient and necessary conditions such that
T
E
∗
(
X
)
{T}_{{E}^{\ast }}\left(X)
is left regular (respectively, right regular, completely regular, intra-regular, and completely simple). Finally, we characterize the
ℱ
{\mathcal{ {\mathcal F} }}
-abundant (respectively,
G
{\mathcal{G}}
-abundant) and left (respectively, right) magnifying elements in
T
E
∗
(
X
)
{T}_{{E}^{\ast }}\left(X)
.