Question

Show that a partially ordered M can have at most one element a such that a <=x for all x in M and at most one element b such that x<=b for all xin M. [If such an a (or b) exists, it is called the least element (greatest element, respectively) of M.]

Answer 1

An element $x\isin M$ is maximal (minimal) if there does not exist $y\isin M\setminus \{x\}$ with

$x\preceq y\ (y\preceq x)$.

So, M can have at most one maximal element and one minimal element.