Filter object (mathematics)
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Informally, filter objects are like filters but ordered the reverse.
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Formal definition of filter objects
Consider the set of filters on some poset (the base poset), further just the set of filters.
Let down is a bijective function from the set of filters to some other set, which is called the set of filter objects. (Its elements are called filter objects.)
We will order filter objects reversely to filters, that is A⊆B ⇔ down A⊇down B for filters A and B.
We will also equate filter objects corresponding to principal filters with the corresponding elements of the base poset.
Reason of introducing filter objects
- The elements of the base poset can be equated with certain filter objects (filter objects corresponding to principal filters).
- The order of filter objects equated with the base poset elements is the same as the order of the base poset.
These properties make in certain situations filter objects more convenient than filters.
Filter objects as objects
Let the function up is the reverse bijection to the function down: up = down-1.
Then any filter object A is defined by the value of up A.
We can describe properties of filter objects with formulas about values of function up for these objects. For example order of filter objects can be described with the formula A⊆B ⇔ up A⊇up B where A and B are filter objects.
It is similar to object oriented programming where objects are defined by properties of their methods.
History of filter object
Filter objects were first introduced by VictorPorton in this article [1].
