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