Let X be a nonempty set, and let

be a family of nonempty subsets of X with the properties that

, and

for all

with

. Let

, and let

. Then

is a subsemigroup of the semigroup

of functions on X having ranges contained in

, where

. For each

, let

be defined by

. Next, we define two congruence relations

and

on

as follows:

and

. We begin this paper by studying the regularity of the quotient semigroups

and

, and the semigroup

. For each

, we see that the equivalence class [

] of

under

is a subsemigroup of

if and only if

is an idempotent element in the full transformation semigroup T(I). Let

,

and

be the sets of functions in

such that

is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups [

],

,

and

of

.