Such a least cut does indeed exist and one has a closure operator on the powerset lattice of all elements.
12.
Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements.
13.
If the closure operator is taken as primitive, the interior operator can be defined as " x"
14.
A set together with a closure operator on it is sometimes called a "'closure system " '.
15.
The convex hull in " n "-dimensional Euclidean space is another example of a finitary closure operator.
16.
Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure.
17.
Every Galois connection ( or residuated mapping ) gives rise to a closure operator ( as is explained in that article ).
18.
The operator \ langle \ rangle is a finitary closure operator on the set of subsets of | \ mathcal A |.
19.
In summary, one can say that every complete lattice is isomorphic to the image of a closure operator on a powerset lattice.
20.
In this way, closure operators and Galois connections are seen to be closely related, each specifying an instance of the other.
