Let E, E₁ denote, respectively, the set of singular idempotents in T[sub]n (the semigroup of all full transformations on a finite set X[sub]n = {1,..., n}) and the set of idempotents of defect 1. For a singular element in Tn, let k(),k₁ () be defined by the properties
∈ Eᵏ⁽ᵃ⁾, ∉ Eᵏ⁽ᵃ⁾⁻¹,
∈ E₁ᵏ¹⁽ᵃ⁾, ∉ E₁ᵏ¹⁽ᵃ⁾⁻¹.
In this Thesis, we obtain results analogous to those of Iwahori (1977), Howie (1980), Saito (1989) and Howie, Lusk and McFadden (1990) concerning the values of k() and k₁() for the partial transformation semigroup P[sub]n. The analogue of Howie and McFadden's (1990) result on the rank of the semigroup K(n,r) = { ∈ T [sub]n: |im | ≤ r,2 ≤ r ≤ n-1} is also obtained.
The nilpotent-generated subsemigroup of P[sub]n was characterised by Sullivan in 1987. In this work, we have obtained its depth and rank.
Nilpotents in IO[sub]n and PO[sub]n (the semigroup of all partial one-one order-preserving maps, and all partial order-preserving maps) are studied. A characterisation of their nilpotent-generated subsemigroups is obtained. So also are their depth and rank. We have also characterised their nilpotent-generated subsemigroup for the infinite set X = {1,2,...}. The rank of the semigroup L(n,r) = {a ∈ S : |im | ≤r, 1 ≤ r ≤ n - 2} is investigated for S = O[sub]n,PO[sub]n,SPO[sub]n and I[sub]n (where O[sub]n is the semigroup of all order-preserving full transformations, SPO[sub]n the semigroup of all strictly partial order- preserving maps, and In the semigroup of one-one partial transformation)
