Graphical Representation of Conjugacy Classes in the Order ? Preserving Partial One ? One Transformation Semigroup

Let X_n={1, 2, 3,. . , n} be the finite set of n objects. The set of all partial transformation of n objects forms a semigroup under the usual composition of functions. It is denoted by T_n, when it is full or total. , I_n, when it is partial one - one and P _n when it is strictly partial. The elements of conjugacy classes of partial one one transformation were studied. The elements in each conjugacy were represented using two line notation. Finally the elements were arranged in their respective conjugacy class with the aids of unlabelled graph. The elements with the same graph structures will contained in the same conjugacy class.