The problem
Consider the following set for divisibility. {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 96}. If 0 is added, the divisibility relation set will no longer be a poset. Please explain why
My Work
From what I've been reading, Division by Zero, I 've come to the conclusion that the divisibility relation on this set is not a poset because it is not reflexive because (0, 0) is not in the relation for divisibility for this relation.
Would the correct justification be that nothing can be divided by zero? This is confusing to me because the definition for a divides b is, is there an integer k such that b = ak? For something like 0 divides 6, this obviously wouldn't be true, because there is no integer k to make 6 = 0(k) true. However for 0 divides 0, any integer k would make 0 = 0(k) true. Can anyone clarify this?