Some transformation semigroup's generators and ranks

Let PTn and Tn denote the semigroups of all partial transformations semigroup and of all transformations semigroup on the natural ordered set X , respectively. In this thesis we give an algorithm to obtain a factorization of the elements of PTn . We use this factorization to obtain information about generating sets for the semigroup of all partial transformations semigroup.Let On denote the semigroup of all order-preserving elements of Tn , and let POn denote the semigroup of all partial order-preserving elements of PTn. . We also use this factorization to obtain information about generating sets for On and POn .