Group closures of injective order-preserving transformations

Given a group G of permutations of a finite n-element set Xn and a transformation f of Xn, the G-closuref : Gof f is the semigroup generated by all the conjugates of f by permutations in G. A semigroup S of transformations of Xn is G-normal if GS = G, where GS consists of all the permutations h of Xn such that h −1 fh∈ S for all f ∈ S.W e may assume that Xn is a chain and we let POIn be the semigroup of all the partial and total one-to-one order preserving transformations of Xn. In the present paper we describe the group Γ = GPOIn, characterize all the inverse Γ-closures of transformations in POIn, and characterize all the inverse Γ-closures that are also Γ-normal.