Monoid is an algebraic structure that consists of a set together with an associative binary operation and an identity element. In order for a set to form a monoid, it must satisfy the following conditions:

1. Closure: For any two elements 'a' and 'b' in the set, their binary operation 'a * b' must also be in the set.

2. Associativity: For any three elements 'a', 'b', and 'c' in the set, the operation must satisfy the associative property, i.e., (a * b) * c = a * (b * c).

3. Identity element: There must exist an element 'e' in the set such that for any element 'a' in the set, the operation 'a * e' and 'e * a' both yield 'a'.

Let's analyze the given options:

a) (a * e) = a
This option represents the existence of an identity element 'e' in the set, which is required for a monoid.

b) (a * e) = (a * c)
This option represents the closure property, as it states that for any element 'a' and 'c' in the set, their binary operation yields the same result.

c) a = (a * (a * e))
This option represents the closure property, as it states that for any element 'a' in the set, the operation 'a * (a * e)' yields the same result as 'a'.

d) (a * e) = a and a = (a * (a * e))
This option represents both the identity element and closure property, as it satisfies conditions (a) and (c) mentioned above.

From the analysis, it is clear that option 'D' satisfies all the conditions required for a monoid, making it the correct answer.