Affiliation:
1. Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University , Pathum Thani , 12120 , Thailand
Abstract
Abstract
Let
P
(
X
)
P\left(X)
be a partial transformation semigroup on a non-empty set
X
X
. For a fixed non-empty subset
Y
Y
of
X
X
, let
P
T
¯
(
X
,
Y
)
=
{
α
∈
P
(
X
)
∣
(
dom
α
∩
Y
)
α
⊆
Y
}
.
\overline{PT}\left(X,Y)=\left\{\alpha \in P\left(X)| \left({\rm{dom}}\hspace{0.33em}\alpha \cap Y)\alpha \subseteq Y\right\}.
Then,
P
T
¯
(
X
,
Y
)
\overline{PT}\left(X,Y)
consists of all the mapping in
P
(
X
)
P\left(X)
that leave
Y
⊆
X
Y\subseteq X
as an invariant. It is a generalization of
P
(
X
)
P\left(X)
since
P
T
¯
(
X
,
X
)
=
P
(
X
)
\overline{PT}\left(X,X)=P\left(X)
. In this article, we present the necessary and sufficient conditions for elements of
P
T
¯
(
X
,
Y
)
\overline{PT}\left(X,Y)
to be regular, left regular, and right regular. The results are used to describe the relationships between these elements and determine their number when
X
X
is a finite set. Moreover, we show that
P
T
¯
(
X
,
Y
)
\overline{PT}\left(X,Y)
is always abundant.