If the second condition is changed to , then is called a dual closure map on .
In the example above, any integer is a fixed point of .
Every closure map can be characterized by an interesting decomposition property: is a closure map iff there is a set and a residuated function such that , where denotes the residual of .
Again, in the example above, , where is the function taking any real number to the largest integer smaller than . is residuated, and its residual is .
Remark. Closure maps are generalizations to closure operator on a set (see the parent entry). Indeed, any closure operator on a set takes a subset of to a subset of satisfying the closure axioms, where Axiom 2 corresponds to condition 2 above, and Axiom 3 says the operator is idempotent. To see that the operator is order preserving, suppose . Then by Axiom 4, and hence . Axiom 1 says that the empty set is a fixed point of the operator. However, in general, this is not the case, for may not even have a minimal element, as indicated by the above example.
- 1 T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005).
|Date of creation||2013-03-22 18:53:55|
|Last modified on||2013-03-22 18:53:55|
|Last modified by||CWoo (3771)|