11. If it does, it is a maximal or greatest element of " S ". 12. Least and greatest elements may fail to exist, as the example of the real numbers shows. 13. However, if it has a greatest element , it can't have any other maximal element. 14. In particular, it contains a least element and a greatest element ( also denoted " universe " ). 15. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal. 16. In a bounded meet-semilattice, the identity 1 is the greatest element of " S ". 17. It should be remarked that the formal definition looks very much like that of a greatest element for an ordered set. 18. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology. 19. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set. 20. The dual notion, the empty lower bound, is the greatest element , top, or unit ( 1 ).
