GATE CS 1997 Q3.1 — Monoids and Groups

To determine the nature of the algebraic structure $(Z, *)$, let us analyze its properties under the operation $n * m = \max(n, m)$ for all $n, m \in \mathbb{Z}$:

1. Closure: For any two integers $n, m \in \mathbb{Z}$, their maximum $\max(n, m)$ is also an integer. Thus, closure holds.
2. Associativity: For any $a, b, c \in \mathbb{Z}$: $$(a * b) * c = \max(\max(a, b), c) = \max(a, b, c)$$ $$a * (b * c) = \max(a, \max(b, c)) = \max(a, b, c)$$ Since $(a * b) * c = a * (b * c)$, the operation is associative. Hence, $(Z, *)$ is a semigroup.
3. Identity Element: An element $e \in \mathbb{Z}$ is an identity element if and only if for all $a \in \mathbb{Z}$: $$a * e = e * a = a \implies \max(a, e) = a$$ For $\max(a, e) = a$ to hold for every integer $a$, we must have $e \le a$ for all $a \in \mathbb{Z}$. Since the set of integers $\mathbb{Z}$ is infinite and has no lower bound, no such integer $e$ exists.

Since there is no identity element in $(Z, *)$, the algebraic structure is not a monoid. Consequently, it cannot be a group or an Abelian group.

Therefore, the correct option is D.