Solution:Analyze each statements:
(A) In any monoid, there exists a unique identity element. This element is such that when combined with any other element in the monoid, it leaves the other element unchanged. So, the statement (A) is correct.
(B) A monoid is indeed a group if there exists an in verse for each element of the monoid. This is one of the defining properties that distinguishes a monoid from a group so, the statement (B) is correct.
(C) A semigroup only needs to have closure and associativeity properties. The property of identity is not necessary for a structure to be considered a semigroup. So, the statement (C) is incorrect.
(D) Quasi- groups do have the closure property. This means that for any two elements in the set, their operation results in another element that is also in the set, So, the statement (D) is correct.