As I went to study the structure of a Lattice I noticed that it consisted on a partially ordered set in which every two elements have a unique supremum and a unique infimum.
Everything until now is okay, but in my teacher textbook it defines supremum/minimum as it follows
An element $c \in A$ is a (upper/lower) bound of a subset $B \subset A$ if there isn't any $b \in B$ which is (subsequent/anteceent) to it. The (supremum/infimum) is the (lowest/highest) of its bounds.
Well, the problem is that if we want to compare the highest or lowest element with other of the set with that definition we can't say that there is any supremum or infimum because any element is subsequent or anteceent to itself.
I interpeted that the infimum/supremum had to be one element in the set $(A-B)$ but as I had said it makes no sense because there couldn't be any Lattice.
Am I interpreting it wrong or it is just that this definition is wrong?