31. Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element . 32. The main property to derive this uniqueness is the following : For every in, is the least element of such that. 33. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. 34. One can define half-closed and closed intervals,, and by adjoining as a least element and / or as a greatest element. 35. By using e " instead of d " in the above definition, one defines the least element of " S ". 36. It is easy to see that 0 is the least element with respect to this order : for all " a ". 37. Here's the general proof : suppose " X " is a nonempty set that you wish to prove has a least element . 38. An example is given by the above divisibility order |, where 1 is the least element since it divides all other numbers. 39. The greatest element in this fiber is the discrete topology on " X " while the least element is the indiscrete topology. 40. An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element .
