I've got the following statement: " If every element of a partially ordered set is maximal, then every element is minimal." My thought was if there is no Relation xRy for every element x of the set, it would also mean that every element must me a minimal since the definition of there being a yRx for an element x of the set would interfere with every element being a maximal.
My example was Set = {2,5,7} with the divisibility relation. Since each element has no other comparable element in the set, they would be both minimal and maximal.
But since there are no relations at all possible is that still a poset or does it interfere with the definition.