21. This set has no greatest element , since for every element of the set, there is another, larger, element. 22. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. 23. The notion of greatest element for a preference preorder would be that of "'most preferred "'choice. 24. Bounded completeness has various relationships to other completeness properties, which are detailed in the article on least and a greatest element . 25. The greatest element in this fiber is the discrete topology on " X " while the least element is the indiscrete topology. 26. Of course, when the restriction of \ leq to S is a total order, the notions of maximal element and greatest element coincide. 27. One can define half-closed and closed intervals,, and by adjoining as a least element and / or as a greatest element . 28. But the notion of " eventually " applies to functions on more general sets, specifically those that have an ordering and no greatest element . 29. :I think you have confused maximal element with greatest element .-- talk ) 19 : 46, 20 February 2008 ( UTC) 30. Dually, the existence of an upper adjoint for " j " is equivalent to " X " having a greatest element .
