complete lattice in a sentence

Examples

1. The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.
2. This order has the desirable property that every subset has a supremum and an infimum : it is a complete lattice.
3. Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element.
4. A complete lattice is "'completely distributive "'if for all such data the following statement holds:
5. Completely distributive complete lattices ( also called " completely distributive lattices " for short ) are indeed highly special structures.
6. When considering directed complete partial orders or complete lattices the additional requirements that the specified suprema exist can of course be dropped.
7. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice.
8. Since the definition also assures the existence of binary meets and joins, complete lattices thus form a special class of bounded lattices.
9. The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either.
10. In summary, one can say that every complete lattice is isomorphic to the image of a closure operator on a powerset lattice.