Now one might still hope that there are some useful cases where the set of generators is sufficiently small for a free complete lattice to exist.
42.
Since the identity function is a closure operator too, this shows that the complete lattices are exactly the images of closure operators on complete lattices.
43.
Since the identity function is a closure operator too, this shows that the complete lattices are exactly the images of closure operators on complete lattices.
44.
Hence one can strengthen the above statement to guarantee that any supremum-preserving map between complete lattices is the lower adjoint of a unique Galois connection.
45.
We2 ve shown that an arbitrary subset of " P " has a supremum, which turns " P " into a complete lattice.
46.
In fact, by considering infinite meets and joins, analogously to the binary meets and joins defined above, one sees that this is a complete lattice.
47.
Likewise, " bounded complete lattice " is almost unambiguous, since one would not state the boundedness property for complete lattices, where it is implied anyway.
48.
Likewise, " bounded complete lattice " is almost unambiguous, since one would not state the boundedness property for complete lattices, where it is implied anyway.
49.
Complete lattices must not be confused with complete partial orders ( " cpo " s ), which constitute a strictly more general class of partially ordered sets.
50.
For an overview of stronger notions of distributivity which are appropriate for complete lattices and which are used to define more special classes of lattices such as distributivity in order theory.
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