Final answer:
The correct answer is D. none of these because in a semigroup, usually several elements can have inverses, not only the identity element. In the context provided, none of the sets - natural numbers, integers, or reals - with their standard operations would fit the criterion given.
Step-by-step explanation:
The student is asking about the characteristics of a mathematical structure known as a semigroup, particularly in regards to an identity element and inverses within the semigroup. A semigroup is an algebraic structure consisting of a set combined with an associative binary operation. The question specifies a semigroup where the only element that has an inverse is the identity element. This is a unique feature, as typically in a group, every element must have an inverse, not just the identity element.
The provided choices seem to refer to sets with binary operations indicated symbolically as parentheses, such as (n, ) or (z, ), which likely represent the sets of natural numbers or integers with a particular operation, respectively. Among these choices, none fit the criterion of a semigroup where only the identity element has an inverse. This is because in semigroups of natural numbers or integers, elements other than the identity element could have inverses depending on the operation used. Therefore, the correct answer to this question is D. none of these.
An example to illustrate this concept using integers: in the semigroup (Z, +), where Z represents the set of all integers and + is addition, every element has an inverse since for any integer a, there exists an integer b = -a such that a + b = 0, which is the identity element for addition. Hence, such a semigroup would not meet the specified criteria. The criteria described are more typical of a group with a single exception for elements other than the identity not having inverses.