Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements.
42.
"Comments " : The Kuratowski closure axioms abstract the properties of the closure operator on a topological space, which assigns to each subset its topological closure.
43.
A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator.
44.
The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span.
45.
*PM : upper set operation is a closure operator, id = 8908 new !-- WP guess : upper set operation is a closure operator-- Status:
46.
*PM : upper set operation is a closure operator, id = 8908 new !-- WP guess : upper set operation is a closure operator-- Status:
47.
This topological closure operator has been generalized in category theory; see " Categorical Closure Operators " by G . Castellini in " Categorical Perspectives ", referenced below.
48.
This topological closure operator has been generalized in category theory; see " Categorical Closure Operators " by G . Castellini in " Categorical Perspectives ", referenced below.
49.
Another representation is obtained by noting that the image of any closure operator on a complete lattice is again a complete lattice ( called its " closure system " ).
50.
Given a Galois connection with lower adjoint and upper adjoint, we can consider the compositions, known as the associated closure operator, and, known as the associated kernel operator.
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