Let (Z, star) be an algebraic structure where Z is the set of integers and the…
1997
Let (Z, star) be an algebraic structure where Z is the set of integers and the operation star is defined by n star m = max(n, m). Which of the following statements is true for (Z, star)?
- A.
(Z, *) is a monoid
- B.
(Z, *) is an Abelian group
- C.
(Z, *) is a group
- D.
None of the above
Attempted by 17 students.
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Correct answer: D
For the operation n * m = max(n, m), closure holds on integers, and the operation is associative and commutative. However, a monoid also requires an identity element.
An identity e would have to satisfy max(n, e) = n for every integer n. This means e <= n for every integer n. No such integer exists, because Z has no least element.
Therefore, (Z, *) is not a monoid. Since every group must first be a monoid and must have an identity, it is neither a group nor an Abelian group.
Hence, none of the listed algebraic-structure classifications is correct.