Affiliation:
1. Department of Mathematics, Research Center in Mathematics and Applied Mathematics, Faculty of Science, Chiang Mai University , Chiang Mai , 50200 , Thailand
Abstract
Abstract
Let
T
(
X
)
T\left(X)
be the full transformation semigroup on a set
X
X
. For an equivalence
E
E
on
X
X
, let
T
E
∗
(
X
)
=
{
α
∈
T
(
X
)
:
∀
x
,
y
∈
X
,
(
x
,
y
)
∈
E
⇔
(
x
α
,
y
α
)
∈
E
}
.
{T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X):\forall x,y\in X,\left(x,y)\in E\iff \left(x\alpha ,y\alpha )\in E\right\}.
For each nonempty subset
Y
Y
of
X
X
, we denote the restriction of
E
E
to
Y
Y
by
E
Y
{E}_{Y}
. Let
T
E
∗
(
X
,
Y
)
{T}_{{E}^{\ast }}\left(X,Y)
be the intersection of the semigroup
T
E
∗
(
X
)
{T}_{{E}^{\ast }}\left(X)
with the semigroup of all transformations with restricted range
Y
Y
under the condition that
∣
X
/
E
∣
=
∣
Y
/
E
Y
∣
| X\hspace{-0.1em}\text{/}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}|
. Equivalently,
T
E
∗
(
X
,
Y
)
=
{
α
∈
T
E
∗
(
X
)
:
X
α
⊆
Y
}
{T}_{{E}^{\ast }}\left(X,Y)=\left\{\alpha \in {T}_{{E}^{\ast }}\left(X):X\alpha \subseteq Y\right\}
, where
∣
X
/
E
∣
=
∣
Y
/
E
Y
∣
| X\hspace{-0.1em}\text{/}\hspace{-0.1em}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}|
. Then
T
E
∗
(
X
,
Y
)
{T}_{{E}^{\ast }}\left(X,Y)
is a subsemigroup of
T
E
∗
(
X
)
{T}_{{E}^{\ast }}\left(X)
. In this paper, we characterize the natural partial order on
T
E
∗
(
X
,
Y
)
{T}_{{E}^{\ast }}\left(X,Y)
. Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of
T
E
∗
(
X
,
Y
)
{T}_{{E}^{\ast }}\left(X,Y)
lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.
Reference13 articles.
1. J. S. V. Symons
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2. J. Sanwong
and
W. Sommanee
, Regularity and Green’s relations on a semigroup of transformation with restricted range, Int. J. Math. Math. Sci. 78 (2008), no. 11, 1–11.
3. K. Sangkhanan
and
J. Sanwong
,
Naturally ordered transformation semigroups with restricted range
,
Proceedings of the 15th Annual Meeting in Mathematics
, Thailand, 2010.
4. L. Sun
and
J. Sun
, A natural partial order on certain semigroups of transformations with restricted range, Semigroup Forum 92 (2016), no. 1, 135–141.
5. H. Pei
, Regularity and Green’s relations for semigroups of transformations that preserve an equivalence, Comm. Algebra 33 (2005), no. 1, 109–118.
Cited by
1 articles.
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