Lesson 3

A minimal element of an order is an element for which no smaller element exists, while a least element is one that is smaller than all other elements. For linear orders, these concepts coincide, but in general, they do not. To give an example, consider the sets \(A\) and \(B\) of points on the plane with coordinate-wise comparison. The order \(A\) has one minimal element, which is also a least one, while in order \(B\), there is no least element at all, but there are infinitely many minimal elements.