Green’s relations and regularity on semigroups of partial transformations with fixed sets

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 byP( X). For a fixed subset Y of X, define a generalization of P( X) byPFix ( X, Y ).= {a ∈ P( X) : ya = y for all y ∈ dom(a) ∩ Y}.In this thesis, we study some well-known and very important concepts in semigroup theory, such as Green’s relations, regularity and abundance on PFix ( X, Y ).