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Given a non-empty set  X  and let  P ( X ) be a partial transformation semigroup on  X . For a fixed subset  Y  of  X , define a generalization of  P ( X ) by In this article, we give a complete description of Green’s relations on  PFix ( X ,  Y ) and apply this to obtain the characterizations of left regular, right regular, intra-regular and completely regular elements of  PFix ( X ,  Y ). The number and the relationship between these elements are also investigated by using finiteness of  X  and

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