On pomonoid of partial transformations of a poset

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.