Green's relations, ideals and regularity on semigroups of partial transformations with invariant set

Let X be a nonempty set. A partial transformation semigroup is the set of all functions from a subset of X into X under the composition of functions, denoted by P(X). For a fixed proper subset Y of X, let PFix(X,Y)={\alpha\in P(X) : y\alpha=y for all y\in\dom\alpha\cap Y}. For \emptyset\neq Y\subseteq X, let \pt=\{\alpha\in P(X) : (\dom\alpha\cap Y)\alpha\subseteq Y\}. Then PFix(X,Y) and \pt are subsemigroups of P(X) and PFix(X,Y)\subseteq\pt. Since \overline{PT}(X,X)=P(X) and PFix(X,\emptyset)=P(X), both semigroups are regarded as extensions of $P(X)$. This thesis explores several well-known and crucial concepts in semigroup theory, including Green's relations, unit-regularity and coregularity in the semigroup \pt. Additionally, we delve into the study of directly finiteness and ideals in both \pt and PFix(X,Y).