Ordering by divisibility

Consider the relation ''\(x\) is a divisor of the number \(y\)'' on the set \(\{1,\ldots,n\}\).

Variant
Prove that the relation is a (weak) order
Solution
Reflexivity: \(x\) is a divisor of itself.

Antisymmetry: if \(x\) is a divisor of \(y\) and \(y\) is a divisor of \(x\), then \(x=y\).

Transitivity: if \(x\) is a divisor of \(y\) and \(y\) is a divisor of \(z\), then \(x\) is a divisor \(z\).
Variant
Does this order have a largest and smallest element?
Solution
The smallest element of the order is 1, and if \(n>2\), then it doesn`t have a largest element. If \(n\leq 2\) then \(n\) is the largest element.
Variant
Does this order have a minimum and maximum element?
Solution
The number 1 is the only minimum element and the elements \(\lfloor\frac{n}2\rfloor+1,…,n\) are the maximum elements.
Variant
What in this order corresponds to an infimum and a supremum of a nonempty subset?
Solution
The infimum is the greatest common divisor and the supremum is the least common multiple.