Let (Z, *) be an algebraic structure where Z is the set of integers and the operation * is defined by n * m = max(n, m)....

Question

Let (Z, *) be an algebraic structure where Z is the set of integers and the operation * is defined by n * m = max(n, m). Which of the following statements is true for (Z, *)? A. (Z, *) is a monoid.  B. (Z, *) is an Abelian group.  C. (Z, *) is a group.  D. None of the above.

Accepted Answer

D. None of the above The operation max is associative and commutative, so (Z, max) is at least a commutative semigroup. For a monoid we need an identity e with max(n, e) = n for all integers n; that would require e to be smaller than or equal to every integer, but Z has no smallest element, so no identity exists. Without an identity it cannot be a monoid, let alone a group, so the answer is None of the above.