Abstract: Let I_X be the symmetric inverse semigroup on the finite set X and E an equivalence on X. Let I_E^*(X)={f∈I_X:for arbitrary elements x,y ∈dom(f),(x,y)∈E if and only if (f(x),f(y))∈E}. Then I_E^*(X) forms an inverse subsemigroup of I_X. If X is a totally ordered set and E is convex equivalence on X,then let OPI_E^*(X) be a semigroup consisting of all preserving orientation partial one-one transformations in I_E^*(X). It is a new class of semigroup and its research is difficult and complicated. By studying it,some new structure and properties of semigroups are explored. Green’s relations on OPI_E^*(X) are described in this paper.

Key words: E^*-preserving, preserving orientation, Green’s relations