Algebraic Structure:
An algebraic structure is a set with one or more operations defined on it, satisfying certain properties. These structures are used to study various mathematical concepts, such as numbers, vectors, and functions.

Semigroup:
A semigroup is an algebraic structure consisting of a set and a binary operation defined on that set. The binary operation combines two elements of the set to produce another element of the set. This operation is generally denoted by '*'.

Option A: (P, *)
In this option, the set is denoted by 'P' and the binary operation is denoted by '*'. This structure satisfies the requirements of a semigroup because:

1. Closure Property: For any two elements 'a' and 'b' in set P, the result of the operation 'a * b' is also an element of set P. Therefore, the operation '* 'is closed on set P.

2. Associativity Property: For any three elements 'a', 'b', and 'c' in set P, the operation '(a * b) * c' is equal to 'a * (b * c)'. In other words, the order in which the operations are performed does not matter.

Since option A satisfies both the closure and associativity properties, it is a valid semigroup.

Option B: (Q, , *)
In this option, the set is denoted by 'Q', and there are two binary operations, denoted by ' ' and '*'. However, a semigroup requires only one binary operation. Therefore, option B cannot be a semigroup.

Option C: (P, )
In this option, the set is denoted by 'P', but there is no binary operation defined. Without a binary operation, it is not possible to define any algebraic structure. Therefore, option C cannot be a semigroup.

Option D: ( , *)
In this option, there is no set defined. A semigroup requires a set on which the binary operation is defined. Therefore, option D cannot be a semigroup.

Conclusion:
The correct answer is option A, (P, *), as it satisfies the requirements of a semigroup by having a set 'P' and a binary operation '*'.