Affiliation:
1. Department of Mathematics and Statistics, College of Science, King Faisal University , P.O. Box 400 , Al-Ahsa , 31982 , Saudi Arabia
Abstract
Abstract
The main objective of this article is to study the ordered partial transformations
PO
(
X
)
{\mathcal{PO}}\left(X)
of a poset
X
X
. The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that
PO
(
X
)
{\mathcal{PO}}\left(X)
is a pomonoid and this pomonoid is denoted by
PO
↑
(
X
)
{{\mathcal{PO}}}^{\uparrow }\left(X)
. Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by
ℐPO
↑
(
X
)
{{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X)
. In case the order on the poset
X
X
is total, we explore some properties of
PO
↑
(
X
)
{{\mathcal{PO}}}^{\uparrow }\left(X)
and
ℐPO
↑
(
X
)
{{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X)
, including regressive, unitary, and reversible.
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