Is it allowed for a poset to have an element that is not related to any other elements?

Question

For example, is the poset $\{(a,b),c\}$ allowed?

Answer 1

No, your set is not a poset. A poset has two parts, the set S and the (partial) order <=, written (S,<=). An antichain is a poset for which no element is related to any other element. The set S = {a,b,c} with the order { (a,a), (b,b), (c,c) } is an antichain. Exercise. Show that antichains can be as large as you want.