[229]Given \(x,y∈ X\) remember that \(x{\lt}y\) means \(x≤ y∧ x≠ y\).
When we have that \(x≤ y\) or \(y≤ x\) we will say that the two elements are ”comparable”. Conversely if neither \(x≤ y\) nor \(y≤ x\) then we will say that the two elements are ”incomparable”.
An element \(m∈ X\) is called maximal if there is no element \(z∈ X\) such that \(m{\lt}z\).
An element \(m∈ X\) is called minimal if there is no element \(z∈ X\) such that \(z{\lt}m\).
An element \(m∈ X\) is called maximum, or greatest element, if, for any element \(z∈ X\), \(z \le m\).
An element \(m∈ X\) is called minimum, or least element, if, for any element \(z∈ X\), \(z \le m\).
Note that the definitions of minimum/minimal can be obtained from maximum/maximal by reversing the order relation (and vice versa).